In fluid dynamics, helicity measures the correlation between velocity and its curl, vorticity, over a spatial volume. Under ‘ideal’ conditions (vanishing viscosity and either homogeneneous density or when pressure may be regarded as a function of density alone), helicity is a topological invariant closely related to the knottedness of vortex lines (Moffatt 1969 J. Fluid Mech. 35 (1), 117–129). Helicity is conserved following a material volume for compact vorticity distributions, i.e. when the vorticity field is tangent to the surface of the volume. There is a related helicity invariant in ideal magnetohydrodynamics involving the correlation between the magnetic potential and its curl, the magnetic field. Helicity is a fragile invariant in the sense that relaxing any one of the ideal conditions results in non-conservation. Unlike energy and enstrophy (mean-square vorticity), helicity is not positive (or sign) definite. Viscous diffusion can create both positive and negative helicity when vortex lines reconnect, something which is topologically forbidden in an ideal fluid where vortex lines move as material curves. Moreover, variable density or more generally compressibility destroys conservation and weakens the association between helicity and vortex-line topology. Furthermore, in compressible flows, the velocity field is not entirely determined from the vorticity field. A recent paper by Boutros & Gibbon (2025) J. Fluid Mech. in this journal explains how one can extend the definition of helicity to control and limit the non-conservation of helicity. This offers a promising way forward in using helicity to characterise flow properties in computational studies of high Reynolds number flows.

The well-known quadratic temperature–velocity (TV) relation is significant for physical understanding and modelling of compressible wall-bounded turbulence. Meanwhile, there is an increasing interest in employing the TV relation for laminar modelling. In this work, we revisit the TV relation for both laminar and turbulent flows, aiming to explain the success of the TV relation where it works, improve its accuracy where it deviates and relax its limitation as a wall model for accurate temperature prediction. We show that the general recovery factor defined by Zhang et al. (J. Fluid. Mech., vol. 739, 2014, pp. 392–440) is not a wall-normal constant in most laminar and turbulent cases. The effective Prandtl number $Pr_e$ is more critical in determining the shape of temperature profiles. The quadratic TV relation systematically deviates for laminar boundary layers irrespective of Mach number and wall boundary conditions. We find a universal distribution of $Pr_e$, based on which the TV relation can be notably improved, especially for cold-wall cases. For turbulent flows, the TV relation as the wall model can effectively improve the near-wall temperature prediction for cold-wall boundary layer cases, but it involves boundary-layer-edge quantities used in the Reynolds analogy scaling, which hinders the application of the wall model in complex flows. We propose a transformation-based temperature wall model by solving inversely the newly developed temperature transformation of Cheng and Fu (Phy. Rev. Fluids, vol. 9, 2024, no. 054610). The dependence on edge quantities is thus removed in the new model and the high accuracy in turbulent temperature prediction is maintained for boundary layer flows.

Starting from the coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures was derived in Fox (2019, J. Fluid Mech. 877, 282). In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients. The ideal-fluid–particle limit of this model is found by letting one of the particle diameters go to zero while the other remains finite. The resulting two-fluid model has closed terms for the spatial fluxes and momentum exchange due to the excluded volume occupied by the particles, e.g. a momentum-exchange term $\boldsymbol {F}_{\!\!fp}$ that depends on gradients of the fluid density $\rho _f$, fluid velocity $\boldsymbol{u}_{f}$ and fluid pressure $p_f$. In Zhang et al. (2006, Phy. Rev. Lett. 97, 048301), the corresponding unclosed momentum-exchange term depends on the divergence of an unknown particle–fluid–particle (pfp) stress (or pressure) tensor. Here, it is shown that the pfp-pressure tensor ${\unicode{x1D64B}}_{\!pfp}$ can be found in closed form from the expression for $\boldsymbol {F}_{\!\!fp}$ derived in Fox (2019, J. Fluid Mech. 877, 282). Remarkably, using this expression for ${\unicode{x1D64B}}_{\!pfp}$ ensures that the two-fluid model for ideal-fluid–particle flow is well posed for all fluid-to-particle material-density ratios $Z = \rho _f / \rho _p$.

We present a mathematical model to investigate heat transfer and mass transport dynamics in the wave-driven free-surface boundary layer of the ocean under the influence of long-crested progressive surface gravity waves. The continuity, momentum and convection–diffusion equations for fluid temperature are solved within a Lagrangian framework. We assume that eddy viscosity and thermometric conductivity are dependent on Lagrangian coordinates, and derive a new form of the second-order Lagrangian mass transport velocity, applicable across the entire finite water depth. We then analyse the convective heat dynamics influenced by the free-surface boundary layer. Rectangular distributions of free-surface temperature (i.e. a Dirichlet boundary condition) are considered, and analytical solutions for thermal boundary layer temperature fields are provided to offer insights into free-surface heat transfer mechanisms affected by ocean waves. Our results suggest the need to improve existing models that neglect the effects of free-surface waves and the free-surface boundary layer on ocean mass transport and heat transfer.

A small sphere fixed at various drafts was subjected to unidirectional broad-banded surface gravity wave groups to investigate nonlinear exciting forces. Testing several incident wave phases and amplitudes permitted the separation of nonlinear terms using phase-based harmonic separation methods and amplitude scaling arguments, which identified third-order forces within the wave frequency range, i.e. third-order first-harmonic forces. A small-body approximation with instantaneous volumetric corrections reproduced the third-order first-harmonic heave forces very well in long waves, and at every tested draft. Further analysis of the numerical model shows these effects are primarily due to instantaneous buoyancy changes, which for a spherical geometry possess a cubic relationship with the wave elevation. These third-order effects may be important for applications such as heaving point absorber wave energy converters, where they reduce the first-harmonic exciting force by ${\sim} 10\, \%$ in energetic operational conditions, an important consideration for power capture.

This work is a numerical study of a transitional shock wave boundary layer interaction (SWBLI). The main goal is to improve our understanding of the well-known low-frequency SWBLI unsteadiness and especially the suspected role of triadic interactions in the underlying physical mechanism. To this end, a direct numerical simulation is performed using a high-order finite-volume scheme equipped with a suitable shock capturing procedure. The resulting database is then extensively post-processed in order to extract the main dynamical features of the interaction zone dynamics (involved characteristic frequencies, characteristics of the vortical structures, etc.). The dynamical organisation and space–time evolution of the flow at dominant frequencies are then further characterised by mean of an spectral proper orthogonal decomposition analysis. In order to study the role of triadic interactions occurring in the interaction region, a bispectral mode decomposition analysis is applied to the database. It allows us to extract the significant triadic interactions, their location and the resulting physical spatial modes. Strong triadic interactions are detected in the downstream part of the separation bubble whose role on the low-frequency unsteadiness is characterised. All the results of the various analyses are then discussed and integrated to formulate a possible mechanism fuelling low-frequency SWBLI unsteadiness.

Geophysical flows are typically composed of wave and mean motions with a wide range of overlapping temporal scales, making separation between the two types of motion in wave-resolving numerical simulations challenging. Lagrangian filtering – whereby a temporal filter is applied in the frame of the flow – is an effective way to overcome this challenge, allowing clean separation of waves from mean flow based on frequency separation in a Lagrangian frame. Previous implementations of Lagrangian filtering have used particle tracking approaches, which are subject to large memory requirements or difficulties with particle clustering. Kafiabad & Vanneste (2023, Computing Lagrangian means, J. Fluid Mech., vol. 960, A36) recently proposed a novel method for finding Lagrangian means without particle tracking by solving a set of partial differential equations alongside the governing equations of the flow. In this work, we adapt the approach of Kafiabad & Vanneste to develop a flexible, on-the-fly, partial differential equation-based method for Lagrangian filtering using arbitrary convolutional filters. We present several different wave–mean decompositions, demonstrating that our Lagrangian methods are capable of recovering a clean wave field from a nonlinear simulation of geostrophic turbulence interacting with Poincaré waves.

Combined surging and pitching of an airfoil at the identical frequency (i.e. synchronously), at four different phase differences, was investigated theoretically and experimentally. The most general unsteady theoretical formulation was adopted to calculate the lift coefficient, and then extended to explicitly compute the unsteady bound vortex sheet. This was used for comparison with experiments and facilitated the computation of both Joukowsky and impulsive-pressure lift contributions. Experiments were performed using a symmetric 18 % thick airfoil in an unsteady wind tunnel at an average Reynolds number of $3.0\times 10^5$, with a free-stream oscillation amplitude of 51 %, an angle-of-attack range of $2^\circ \pm 2^\circ$ and a reduced frequency of 0.097. In general, excellent correspondence was observed between theory and experiment, representing the first direct experimental validation of the general theory. It was shown, both theoretically and experimentally, that the lift coefficient was not accurately represented by independent superposition of surging and pitching effects, due to variations in the instantaneous effective reduced frequency not accounted for during pure pitching. Deviations from theory, observed at angle-of-attack phase leads of $90^\circ$ and $180^\circ$, were attributed to bursting of separation bubbles during the early stages of the acceleration phase. The largest deviations occurred when the impulsive-pressure lift contribution was small relative to the Joukowsky contribution, because the latter was most affected by bubble bursting. Bubble bursting resulted in large form-drag oscillations that occurred at identical phase angles within the oscillation cycle, irrespective of the phase difference between surging and pitching, as well as in the absence of pitching.

The orientational trajectories of rod-like particles suspended in a liquid are influenced by their surroundings, such as the type of flow and nearby walls, and deviate from the well-known Jeffery orbits in shear flows. We consider two types of shear flows between two parallel planar walls: wall-driven simple shear flow (C-flow), and parabolic flow driven by an external body force (P-flow). We simulated hydrodynamically interacting rod-like particles using a chain-of-spheres model immersed in a lattice Boltzmann fluid within a confined channel. As these particles in shear flows approach the wall, their orbits become flattened, exhibiting a ‘swinging motion’ on a plane parallel to the wall. Near the wall, the influence of the wall on the orbital motion varies depending on the flow type. In P-flow, the particles maintain their periodic swinging motions, whereas in C-flow, they stop swinging and align with the flow direction. This difference arises due to distinct hydrodynamic interactions with the wall in each flow type. Simulations also replicated the ‘pole-vaulting’ motion, where particles move away from the wall during their tumbling motion. For weakly sedimenting particles under shear flows, both flow types showed behaviour similar to that of neutrally buoyant particles. However, in P-flow, driven by gravity towards the wall, the particles cease their swinging motion and align perpendicularly to the flow direction, consistent with experimental observations.

There is a reasonable possibility that the present-day Atlantic Meridional Overturning Circulation is in a bistable regime, hence it is relevant to compute pathways of noise-induced transitions between the stable equilibrium states. Here, the most probable transition pathway of a noise-induced tipping of the northern overturning circulation in a spatially-continuous two-dimensional model with surface temperature and stochastic salinity forcings is computed directly using large deviation theory. This pathway reveals the fluid dynamical mechanisms of such a tipping. Paradoxically it starts off with a strengthening of the northern overturning circulation before a short but strong salinity pulse induces a second overturning cell. The increased atmospheric energy input of this two-cell configuration cannot be mixed away quickly enough, leading to the collapse of the northern overturning cell, and finally resulting in a southern overturning circulation. Additionally, the approach allows us to compare the probability of this transition under different parameters in the deterministic part of the salinity surface forcing, which quantifies the increase in transition probability as the bifurcation point of the system is approached.

The shallow-water equations are widely used to model interactions between horizontal shear flows and (rotating) gravity waves in thin planetary atmospheres. Their extension to allow for interactions with magnetic fields – the equations of shallow-water magnetohydrodynamics (SWMHD) – is often used to model waves and instabilities in thin stratified layers in stellar and planetary atmospheres, in the perfectly conducting limit. Here we consider how magnetic diffusion should be added to the equations of SWMHD. This is crucial for an accurate balance between advection and diffusion in the induction equation, and hence for modelling instabilities and turbulence. For the straightforward choice of Laplacian diffusion, we explain how fundamental mathematical and physical inconsistencies arise in the equations of SWMHD, and show that unphysical dynamo action can result. We then derive a physically consistent magnetic diffusion term by performing an asymptotic analysis of the three-dimensional equations of magnetohydrodynamics in the thin-layer limit, giving the resulting diffusion term explicitly in both planar and spherical coordinates. We show how this magnetic diffusion term, which allows for a horizontally varying diffusivity, is consistent with the standard shallow-water solenoidal constraint, and leads to negative semidefinite Ohmic dissipation. We also establish a basic type of antidynamo theorem.

We examine the separate effects of turbulence beneath a free surface and non–breaking surface capillary waves on the gas-transfer velocity of atmospheric oxygen into water across an air–water interface. The experiments are conducted in a recirculating open water channel with quiescent air, where atmospheric oxygen naturally dissolves into the water via the exposed surface. Through the combination of an active turbulence grid and an array of surface penetrating dowels, we are able to separate the effects of sub-surface turbulence and surface capillary waves. The findings demonstrate that the gas-transfer velocity trends with the turbulence properties, not the capillary wave properties, thus indicating that, when both are present, it is the sub-surface turbulence, not the capillary waves, that plays the dominant role in determining the rate of gas transfer across an air–water interface in the non-breaking capillary wave regime.

Laminar–turbulent transition on the suction surface of the LM45.3p blade ($20\,\%$ thickness) was investigated using wall-resolved large eddy simulation (LES) at a chord Reynolds number of $Re_c=10^6$ and angle of attack $4.6^\circ$. The effects of anisotropic free stream turbulence (FST) with intensities $TI=0\,\%$–$7\,\%$ were examined, with integral length scales scaled down from atmospheric measurements. At $TI=0\,\%$, a laminar separation bubble (LSB) forms and transition is initiated by Kelvin–Helmholtz vortices. At low FST levels ($0\,\%\lt TI \leqslant 2.4\,\%$), robust streak growth via the lift-up mechanism suppresses the LSB, while transition dynamics shifts from two-dimensional Tollmien–Schlichting (TS) waves ($TI=0.6\,\%$) to predominantly varicose inner and outer instabilities ($TI=1.2\,\%$ and $2.4\,\%$) induced by the wall-normal shear and inflectional velocity profiles. The critical disturbance kinetic energy scales with $TI^{-1.80\pm 0.11}$, compared with $TI^{-2.40}$ from Mack’s correlation. For $TI\geqslant 4.5\,\%$, bypass transition dominates, driven by high-frequency boundary layer perturbations and streak breakdown via outer sinuous modes induced by the spanwise shear and inflectional velocity profiles. The scaling of streak amplitudes with $TI$ becomes sub-linear and spanwise non-uniformity characterises the turbulent breakdown. The critical disturbance kinetic energy reduces to $TI^{-0.90\pm 0.16}$, marking a transition regime distinct from modal mechanisms. The onset of bypass transition ($TI\approx 2.4\,\%{-}4.5\,\%$) aligns with prior studies of separated and flat-plate flows. A proposed turbulence spectrum cutoff links atmospheric measurements to wind tunnel data and Mack’s correlation, offering a framework for effective $TI$ estimation in practical environments.

We address the problem of shock-induced ignition and transition to detonation in a reactive medium in the presence of mechanically induced fluctuations by a moving oscillating piston. For the inert problem prior to ignition, we provide a novel closed-form model in Lagrangian coordinates for the generation of the train of compression and expansions, their steepening into a train of N-shock waves and their reflection on the lead shock, as well as the distribution of the energy dissipation rate in the induction zone. The model is found to be in excellent agreement with numerics. Reactive calculations were performed for hydrogen and ethylene fuels using a novel high-fidelity scheme to solve the reactive Euler equations written in Lagrangian coordinates. Different regimes of ignition and transition to detonation, controlled by the time scale of the forcing and the two time scales of the chemistry: the induction and reaction times. Two novel hotspot cascade mechanisms were identified. The first relies on the coherence between the sequence of hotspot formation set by the piston forcing and forward-wave interaction with the lead shock, generalising the classic runaway in fast flames. The second hotspot cascade is triggered by the feedback between the pressure pulse generated by the first-generation hotspot cascade and the shock. For slow forcing, the sensitisation is through a modification to the classic runaway process, while the high-frequency regime leads to very localised subcritical hotspot formation controlled by the cumulative energy dissipation of the first-generation shocks at a distance comparable with the shock formation location.

The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting, the conservation law holds only under the assumption that the pressure is barotropic. Let us consider a volume $V$ containing a compressible fluid with density $\rho$, velocity field $\textbf{u}$ and vorticity $\boldsymbol{\omega}$. We show that by introducing a new definition of helicity density $h_{\rho }=(\rho {\boldsymbol {u}})\cdot \mbox {curl}\,(\rho {\boldsymbol {u}})$ the barotropic assumption on the pressure can be removed, although ${\int _{V}} h_{\rho }{\rm d}V$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_{\rho }$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux ${\boldsymbol {J}}_{\rho }$ contains all the pressure terms and whose source involves the potential vorticity $q = \boldsymbol{\omega} \cdot \nabla \rho$. Therefore, the rate of change of ${\int _{V}} h_{\rho }{\rm d}V$ no longer depends on the pressure and is easier to analyse, as it depends only on the potential vorticity and kinetic energy as well as $\mbox {div}\,{\boldsymbol {u}}$. This result also carries over to the inhomogeneous incompressible Euler equations for which the potential vorticity $q$ is a material constant. Therefore, $q$ is bounded by its initial value $q_{0}=q({\boldsymbol {x}},\,0)$, which enables us to define an inverse resolution length scale $\lambda _{H}^{-1}$ whose upper bound is found to be proportional to $\|q_{0}\|_{\infty }^{2/7}$. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.

Gravity currents are a ubiquitous density-driven flow occurring in both the natural environment and in industry. They include: seafloor turbidity currents, primary vectors of sediment, nutrient and pollutant transport; cold fronts; and hazardous gas spills. However, while the energetics are critical for their evolution and particle suspension, they are included in system-scale models only crudely, so we cannot yet predict and explain the dynamics and run-out of such real-world flows. Herein, a novel depth-averaged framework is developed to capture the evolution of volume, concentration, momentum and turbulent kinetic energy from direct integrals of the full governing equations. For the first time, we show the connection between the vertical profiles, the evolution of the depth-averaged flow and the energetics. The viscous dissipation of mean-flow energy near the bed makes a leading-order contribution, and an energetic approach to entrainment captures detrainment of fluid through particle settling. These observations allow a reconsideration of particle suspension, advancing over 50 years of research. We find that the new formulation can describe the full evolution of a shallow dilute current, with the accuracy depending primarily on closures for the profiles and source terms. Critically, this enables accurate and computationally efficient hazard risk analysis and earth surface modelling.

We investigate the momentum fluxes between a turbulent air boundary layer and a growing–breaking wave field by solving the air–water two-phase Navier–Stokes equations through direct numerical simulations. A fully developed turbulent airflow drives the growth of a narrowbanded wave field, whose amplitude increases until reaching breaking conditions. The breaking events result in a loss of wave energy, transferred to the water column, followed by renewed growth under wind forcing. We revisit the momentum flux analysis in a high-wind-speed regime, characterized by the ratio of the friction velocity to wave speed $u_\ast /c$ in the range $[0.3\,{-}\,0.9]$, through the lens of growing–breaking cycles. The total momentum flux across the interface is dominated by pressure, which increases with $u_\ast /c$ during growth and reduces sharply during breaking. Drag reduction during breaking is linked to airflow separation, a sudden acceleration of the flow, an upward shift of the mean streamwise velocity profile and a reduction in Reynolds shear stress. We characterize the reduction of pressure stress and flow acceleration through an aerodynamic drag coefficient by splitting the analysis between growing and breaking stages, treating them as separate subprocesses. While drag increases with $u_\ast /c$ during growth, it decreases during breaking. Averaging over both stages leads to a saturation of the drag coefficient at high $u_\ast /c$, comparable to what is observed at high wind speeds in laboratory and field conditions. Our analysis suggests that this saturation is controlled by breaking dynamics.

Dispersion in spatio-temporal random flows is dominated by the competition between spatial and temporal velocity resets along particle paths. This competition admits a range of normal and anomalous dispersion behaviours characterised by the Kubo number, which compares the relative strength of spatial and temporal velocity resets. To shed light on these behaviours, we develop a Lagrangian stochastic approach for particle motion in spatio-temporally fluctuating flow fields. For space–time separable flows, particle motion is mapped onto a continuous time random walk (CTRW) for steady flow in warped time, which enables the upscaling and prediction of the large-scale dispersion behaviour. For non-separable flows, we measure Lagrangian velocities in terms of a new sampling variable, the average number of velocity transitions (both temporal and spatial) along pathlines, which renders the velocity series Markovian. Based on this, we derive a Lagrangian stochastic model that represents particle motion as a coupled space–time random walk, that is, a CTRW for which the space and time increments are intrinsically coupled. This approach sheds light on the fundamental mechanisms of particle motion in space–time variable flows, and allows for its systematic quantification. Furthermore, these results indicate that alternative strategies for the analysis of Lagrangian velocity data using new sampling variables may facilitate the identification of (hidden) Markov models, and enable the development of reduced-order models for otherwise complex particle dynamics.