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.

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.

We propose an analytical approach based on the Frenet–Serret (FL) frame field, where an FL frame and the corresponding curvature and torsion are defined at each point along magnetic field lines, to investigate the evolution of magnetic tubes and their interaction with vortex tubes in magnetohydrodynamics. Within this framework, simplified expressions for the Lorentz force, its curl, the dynamics of flux tubes and helicity are derived. We further perform direct numerical simulations on the linkage between the magnetic and vortex tubes and investigate the effect of the initial angle $\theta$, ranging from $0^{\,\circ}$ to $45^{\,\circ}$, on their evolution. Our results show that magnetic tubes with non-zero curvature generate Lorentz forces, which in turn produce dipole vortices. These dipole vortices lead to the splitting of the magnetic tubes into smaller structures, releasing magnetic energy. Both magnetic and vortex tubes exhibit quasi-Lagrangian behaviour, maintaining similar shapes during initial evolution and consistent relative positions over time. A vortex tube with strength comparable to that of the magnetic tube, where the kinetic energy induced by the vortex tube is of the same order as the magnetic energy in the magnetic tube, can inhibit magnetic tube splitting by disrupting the formation of vortex dipoles. Additionally, minor variations in the angular configuration of the vortex tubes significantly influence their interaction with the magnetic field and the evolution of large-scale flow structures.

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.

The reduction of the hydrodynamic forces exerted on a bluff body in an incoming flow has been an issue of interest in fluid mechanics for many years. However, the Magnus effect indicates possible drag reduction but with the lift being increased significantly. This study is aimed at the simultaneous lift and drag reduction for which we consider a constant incoming flow past a circular cylinder or a sphere in the $x$-direction. Force element analysis (FEA) indicates the possibility of reducing the drag exerted on a circular cylinder or a sphere by rotating (say, clockwise about the $z$-axis) only the front half of the circular cylinder or the sphere. More precisely, we rotate the object but with the rear half covered by a closely spaced hood. Numerical simulations show that by increasing the dimensionless rotational speed $\alpha$: (i) the flow can be quickly stabilised to a steady state; (ii) the mean drag steadily decreases to zero and then becomes negative as $\alpha$ is further increased across the critical $\alpha _I = 4.11$ for the circular cylinder at $Re$ = 200, $\alpha _I = 4.81$ for the sphere at $Re$ = 200 and $\alpha _I = 4.92$ for the sphere at $Re$ = 300; (iii) the mean value of the lift decreases from zero to negative and then increases beyond zero, and in addition, the amplitude of the lift gradually decreases for the circular cylinder; the mean value of the lift decreases from zero to negative for the sphere; (iv) the side force is almost zero – the flow over the sphere is plane-symmetric about the $x{-}y$ plane. These features are compared with the flow past a rotating circular cylinder or a rotating sphere (Magnus effect). Notably, there is a range of flows that can be of practical use for: (a) the circular cylinder where the drag is greatly reduced while the lift is small in magnitude and (b) the sphere where the drag is greatly reduced while the lift is negative in magnitude and the side force is close to 0.

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 investigate the statistical properties of kinetic and thermal dissipation rates in two-dimensional/three-dimensional vertical convection of liquid metal ($Pr = 0.032$) within a square cavity. Two situations are specifically discussed: (i) classical vertical convection with no external forces and (ii) vertical magnetoconvection with a horizontal magnetic field. Through an analysis of dissipation fields and a reasonable approximation of buoyancy potential energy sourced from vertical heat flux, the issue of the ‘non-closure of the dissipation balance relation’, which has hindered the application of the GL theory in vertical convection, is partially resolved. The resulting asymptotic power laws are consistent with existing laminar scaling theories and even show certain advantages in validating simulations with large Prandtl number ($Pr$). Additionally, a full-parameter model and prefactors applicable to low-$Pr$ fluids are provided. The extension to magnetoconvection naturally introduces the approximate expression for total buoyancy potential energy and necessitates adjustments to the contributions of kinetic dissipation in both the bulk and boundary layer. The flow dimensionality and boundary layer thickness are key considerations in this analysis. The comprehension of Joule dissipation has been updated: the Lorentz force generates positive dissipation in the bulk by suppressing convection, while in the Hartmann layer, shaping the exponential boundary layer requires the fluid to perform positive work to accelerate, leading to negative dissipation. Finally, the proposed transport equations for magnetoconvection are supported by current direct numerical simulation (DNS) and literature data, and the applicability of the model is discussed.

This study investigates the hydrodynamic interaction between a fully submerged buoyant pendulum and surface gravity waves, focusing on its primary and subharmonic resonance behaviour. The oscillatory motion of the pendulum is driven by fluid drag, with primary resonance occurring at the forcing frequency (viz. the wave frequency) and subharmonic resonance manifesting at half the forcing frequency. Both resonances exhibit nonlinear characteristics, including jump-up, jump-down phenomena and hysteresis. Furthermore, particle image velocimetry results reveal that the velocity fields of the surrounding fluid oscillate at the forcing frequency, confirming that subharmonic resonance is not induced by subharmonic excitation within the velocity field. Experimental observations are validated through both analytical and numerical methods, particularly within the primary and subharmonic resonance frequency ranges. The theoretical model describes the transverse motion of the pendulum using a nonlinear ordinary differential equation, with the method of multiple scales employed for the analytical solution. These analyses reveal the nonlinear characteristics of the system, e.g. bistable response of the primary/subharmonic resonances, and identify three distinct response regions based on the forcing frequency and amplitude. The system exhibits primary resonance regardless of the excitation strength; however, an unstable solution arises if the excitation level surpasses a specific threshold value. In contrast, subharmonic resonance is triggered only when the excitation amplitude exceeds a critical value. Furthermore, the experimental hysteresis curve confirms the theoretically predicted primary and subharmonic resonances, along with the jump-up and jump-down characteristics.

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.

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.

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.

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.

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.

The Hasselmann equation for the nonlinear interactions of deep-water gravity waves differs from other four-wave kinetic equations by the interaction coefficient. The explicit formula for this coefficient (e.g. Krasitskii, J. Fluid. Mech., vol. 272, 1994, pp. 1–20; Zakharov, Eur. J. Mech. B/Fluids, vol. 18. issue 3, 1999, pp. 327–344) is of great complexity and leaves its properties obscured. We provide analytical results for the behaviour of the coefficient in different domains. The Phillips curve and discrete interaction approximation-like quadruplets are studied in detail. The coupling coefficient for the long–short wave interactions is calculated and found to be surprisingly small. This smallness greatly reduces the non-locality of the interactions.

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.

We derive a depth-averaged model consistent with the $\mu (I)$ rheology for an incompressible granular flow down an inclined plane. The first two variables of the model are the depth and the depth-averaged velocity. The shear is also taken into account via a third variable called enstrophy. The obtained system is a hyperbolic system of conservation laws, with an additional equation for the energy. The system is derived from an asymptotic expansion of the flow variables in powers of the shallow-water parameter. This method ensures that the model is fully consistent with the rheology. The velocity profile is a Bagnold profile at leading order and the first-order correction to this profile can be calculated for flows that are not steady uniform. The first-order correction to the classical granular friction law is also consistently written. As a consequence, the instability threshold of the steady uniform flow is the same for the depth-averaged model and for the governing equations. In addition, a higher-order version that contains diffusive terms is also presented. The spatial growth rate, the phase velocity and the cutoff frequency of the version with diffusion are in good agreement with the experimental data and with the theoretical predictions for the rheology. The mathematical structure of the equations enables us to use well-known and stable numerical solvers. Numerical simulations of granular roll waves are presented. The model has the same limitations as the $\mu (I)$ rheology, in particular for the solid/ liquid and liquid/gas transitions, and needs therefore a regularisation for these transitions.

The formation of Kelvin–Helmholtz-like rollers (referred to as K–H rollers) over riblet surfaces has been linked to the drag-increasing behaviour seen in certain riblet geometries, such as sawtooth and blade riblets, when the riblet size reaches sufficiently large viscous scales (Endrikat et al. (2021a), J. Fluid Mech. 913, A37). In this study, we focus on the sawtooth geometry of fixed physical size, and experimentally examine the response of these K–H rollers to further increases in viscous scaled riblet sizes, by adopting the conventional approach of increasing freestream speeds (and consequently, the friction Reynolds number). Rather than continual strengthening, the present study shows a gradual weakening of these K–H rollers with increasing sawtooth riblet size. This is achieved by an analysis of the roller geometric characteristics using both direct numerical simulations and hot-wire anemometry databases at matched viscous scaled riblet spacings, with the former used to develop a novel methodology for detecting these rollers via streamwise velocity signatures (e.g. as acquired by hot wires). Spectral analysis of the streamwise velocity time series, acquired within riblet grooves, reveals that the frequencies (and the streamwise wavelengths) of the K–H rollers increase with increasing riblet size. Cross-correlation spectra, estimated from unique two-point hot-wire measurements in the cross-plane, show a weakening of the K–H rollers and a reduction in their wall-normal coherence with increasing riblet size. Besides contributing to our understanding of the riblet drag-increasing mechanisms, the present findings also have implications for the heat transfer enhancing capabilities of sawtooth riblets, which have been associated previously with the formation of K–H rollers. The present study also suggests conducting future investigations by decoupling the effects of viscous scaled riblet spacing and friction Reynolds numbers, to characterise their influence on the K–H rollers independently.

We present direct numerical simulations of a supersonic, zero-pressure-gradient, adiabatic turbulent boundary layer at a free-stream Mach number of $M_\infty =2$, over cubical roughness elements. The simulations are complemented by a subsonic rough-wall boundary layer over the same geometry, alongside reference smooth-wall data, allowing us to elucidate compressibility effects. The simulations feature turbulent flow transitioning from a smooth to a rough surface with an extended computational domain to facilitate recovery. At the smooth-to-rough transition, we compare the development of an internal boundary layer between the subsonic and supersonic cases, introducing a novel definition of its height that is less sensitive to local compressibility effects. We demonstrate that, although the internal boundary-layer growth is similar to the subsonic case, a delayed equilibrium is expected for the supersonic case due to the sudden growth of the external boundary-layer thickness at the onset of roughness. Turbulence statistics are then evaluated far from the surface transition, where various compressibility transformations reveal outer-layer similarity for the mean velocity. We find that the classical van Driest II transformation can also be applied to rough walls, at least in the adiabatic case. Analysis of thermal statistics for the supersonic case confirms the significant influence that roughness has on both mean and fluctuating temperature fields, which, unlike velocity fields, do not display outer-layer similarity. Nonetheless, we find that the temperature–velocity relation established for smooth walls is also valid over rough surfaces, implying that the mean temperature field can be predicted solely based on the mean velocity.