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.

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.

The hydrodynamic forces acting on an undulating swimming fish consist of two components: a drag-based resistive force, and a reactive force originating from the necessary acceleration of an added mass of water. Lighthill’s elongated-body theory, based on potential flow, provides a framework for calculating this reactive force. By leveraging the high aspect ratio of most fish, the theory simplifies the problem into a series of independent two-dimensional slices of fluids along the fish’s body, which exchange momentum with the body and neighbouring slices. Using momentum conservation arguments, Lighthill’s theory predicts the total thrust generated by an undulating fish, based solely on the dimensions and kinematics of its caudal fin. However, the assumption of independent slices has led to the common misconception that the flow produced lacks a longitudinal component. In this paper, we revisit Lighthill’s theory, offering a modern reinterpretation using essential singularities of potential flows. We then extend it to predict the full three-dimensional flow field induced by the fish’s body motion. Our results compare favourably with numerical simulations of realistic fish geometries.

The combined effects of the imposed vertical mean magnetic field ($B_0$, scaled as the Alfvèn velocity) and rotation on the heat transfer phenomenon driven by the Rayleigh–Taylor (RT) instability are investigated using direct numerical simulations. In the hydrodynamic (HD) case, as the strength of the Coriolis frequency ($f$) increases, the Coriolis force enhances the mixing of fluids that dampens the growth of the mixing layer height ($h$) and reversible exchanges between the fluids, leading to a reduction in the heat transport, characterised by the Nusselt number ($Nu$). In non-rotating magnetohydrodynamic (MHD) cases, we find a significant delay in the onset of RT instability with increasing $B_0$, consistent with the linear theory in the literature. The imposed $B_0$ forms vertically elongated thermal plumes that exhibit a larger reversible buoyancy flux due to limited mixing, enabling them to transport heat efficiently between the bottom hot fluid and the upper cold fluid. This leads to enhanced heat transfer in the initial regime of unbroken elongated plumes in non-rotating MHD cases compared to the corresponding HD case. In the turbulent regime of broken small-scale structures, the imposed $B_0$ collimates the flow along the vertical magnetic field lines, reducing vertical velocity fluctuations ($u_3^{\prime }$) and increasing the growth of $h$. The increased $h$ primarily drives the heat transfer enhancement in the turbulent regime of non-rotating MHD over the corresponding HD case. When rotation is added along with the imposed $B_0$, the growth and breakdown of vertically elongated plumes are inhibited by the instability-damping effect of the Coriolis force. Consequently, heat transfer is also reduced in the rotating MHD cases compared to the corresponding non-rotating MHD cases. Interestingly, heat transport in rotating MHD cases is enhanced compared to the corresponding rotating HD cases because $B_0$ reduces mixing and mitigates the instability-damping effect of the Coriolis force. The presence of the ultimate state regime $Nu\simeq Ra^{1/2}Pr^{1/2}$, where $Ra$ is the Rayleigh number and $Pr$ is the Prandtl number, is observed in the non-rotating HD and MHD cases. However, the rotating HD and MHD cases depart from this ultimate state scaling. Furthermore, the dynamic balance between different forces is analysed to understand the behaviour of the thermal plumes. The turbulent kinetic energy budget reveals the conversion of the turbulent kinetic energy, generated by the buoyancy flux, into turbulent magnetic energy.

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.

To analyse compressibility-induced non-Oberbeck–Boussinesq (NOB-II) effects, we present a lattice Boltzmann (LB) model capable of simulating supercritical fluids. The LB model is validated using analytical solutions and experimental data. Using this model, we conduct two-dimensional laminar LB simulations of Rayleigh–Bénard convection (RBC) in supercritical fluids. Our results reveal that the ratio of the adiabatic temperature difference to the total temperature difference, $\alpha$, effectively indicates the intensity of NOB-II effects. We find that, NOB-II effects do not break the symmetry of the temperature, density or momentum fields. However, due to density differences between the upper and lower regions, NOB-II effects break the velocity symmetry. Moreover, we report for the first time the density inversion phenomenon in RBC, wherein convection can still occur when the bottom fluid is denser than the top fluid. The condition for density inversion is given as $\alpha \gt (c_p - c_v)/{c_p}$, where $c_p$ and $c_v$ are the specific heat capacities at constant pressure and volume, respectively. This inversion is attributed to the coupling effect of a significant pressure gradient and fluid compressibility. Our results also show that for a given Rayleigh number, NOB-II effects have no impact on the Reynolds number. However, as $\alpha$ approaches 1, the Nusselt number decreases linearly towards 1, indicating significant heat transfer deterioration (HTD). The mechanism underlying HTD is attributed to the compression work term in the energy equation, which absorbs heat from the hot plume in central region, diminishing its capacity to transfer heat from the bottom to the top plate.