The flow past a $6:1$ prolate spheroid at a moderate pitch angle $\alpha =10^\circ$ is investigated with a focus on the turbulent wake in a high-fidelity large eddy simulation (LES) study. Two length-based Reynolds numbers, ${\textit{Re}}_L=3\times 10^4$ and $9\times 10^4$, and four Froude numbers, ${\textit{Fr}} = \infty \text{(unstratified)}, 6, 1.9 \text{ and }1$, are selected for the parametric study. Spectral proper orthogonal decomposition (SPOD) analysis of the flow reveals the leading coherent modes in the unsteady separated flow at the tail of the body. At the higher ${\textit{Re}}_L=9\times 10^4$, a high-frequency spanwise flapping of shear layers on either side of the body is observed in the separated boundary layer for all cases. The flapping does not perturb the lateral symmetry of the wake. At ${\textit{Fr}}=\infty$, a low-frequency oscillating laterally asymmetric mode, which is found in addition to the shear-layer mode, leads to a sidewise unsteady lateral load. All temporally averaged wakes at ${\textit{Re}}=9\times 10^4$ are found to be spanwise symmetric in the mean as opposed to the lower ${\textit{Re}}=3\times 10^4$, at which the ${\textit{Fr}}=\infty \text{ and }6$ wakes exhibit asymmetry. The turbulent kinetic energy (TKE) budget is compared among cases. Here, ${\textit{Fr}}=\infty$ exhibits higher production and dissipation compared with ${\textit{Fr}}=6 \text{ and }1.9$. The streamwise vortex pair in the wake induces a significant mean vertical velocity ($U_z$). Therefore, in contrast to straight-on flow, the terms involving gradients of $U_z$ matter to TKE production. Buoyancy reduces $U_z$ and also the Reynolds shear stresses involving $u^{\prime}_z$. Through this indirect mechanism, buoyancy exerts control on the wake TKE budget, albeit being small relative to production and dissipation. Buoyancy, through the baroclinic torque, is found to qualitatively affect the streamwise vorticity. In particular, the primary vortex pair is extinguished in the intermediate wake and two new vortex pairs form with opposite-sense circulation relative to the primary.

This paper investigates the transient characteristics of uniform momentum zones (UMZs) in a rapidly accelerating turbulent pipe flow using direct numerical simulation datasets starting from an initial friction Reynolds number ($Re_{\tau 0}) = 500$ up to a final friction Reynolds number ($Re_{\tau 1}) = 670$. Instantaneous UMZs are identified following the identification methodology proposed by Adrian et al. (2000 J. Fluid Mech. vol. 422, pp. 1–54). The present results reveal that, as the flow rapidly accelerates, the average number of UMZs drops. However, as the flow recovers, it is regained. This result is complemented by the temporal evolution of the average number of internal shear layers. The temporal evolution of UMZs reveals that UMZs sustain their hierarchical flow arrangement with slower zones near the wall and faster zones away from the wall throughout the rapid turbulent flow acceleration. The results show that UMZs speed up during the inertial and pre-transition phases, and progressively slow down during the transition and core-relaxation stages. It is also revealed that UMZs near the wall respond first to flow instability and show earlier signs of recovery based on UMZ kinematic results. Finally, the dominant quadrant behaviour of Reynolds shear stress within UMZs has been investigated. It is found that, prior to the flow excursion, the UMZs nearest to the wall are always $Q2$ dominated, while the rest of the UMZs are always $Q4$ dominated. This behaviour is detected to not change during and after the flow excursion, suggesting that this is a characteristic behaviour of UMZs in accelerating turbulent wall-bounded flows.

Magneto-gravity-precessional instability, which results from the excitation of resonant magneto-inertia-gravity (MIG) waves by a background shear generic to precessional flows, is addressed here. Two simple background precession flows, that of Kerswell (1993 Geophys. Astrophys. Fluid Dyn. vol 72, no. 1–4, pp. 107–144), and that of Mahalov (1993 Phys. Fluids A: Fluid Dyn. vol. 5, no. 4, pp. 891–900), are considered. We analytically perform an asymptotic analysis to order ${ O}(\varepsilon ),$ where $\varepsilon$ denotes the Poincaré number, i.e. the precession parameter, and determine the maximum growth rate of the destabilizing subharmonic resonances of MIG waves: that between two fast modes, that between two slow modes and that between a fast mode and a slow mode (mixed modes). The domains of the $(K_0 B_0/\varOmega _0, N/\varOmega _0)\hbox{-}$plane for which this instability operates are identified, where $1/K_0$ denotes a characteristic length scale, $B_0$ is the unperturbed Alfvén velocity, $\varOmega _0$ is the rotation rate and $N$ denotes the Brunt–Väisälä frequency. We demonstrate that the $N\rightarrow 0$ limit is, in fact, singular (discontinuous). At large $K_0B_0/\varOmega _0,$ stable stratification acts to suppress the destabilizing resonance between two fast modes as well as that between two slow modes, whereas it revives the destabilizing resonance between a fast mode and a slow mode provided $N\lt \varOmega _0,$ because, without stratification, the maximal growth rate of this instability approaches zero as $K_0B_0/\varOmega _0\rightarrow +\infty .$ This would be relevant for the generation of the mean electromotive force, and hence, the $\alpha \hbox{-}$effect in helical magnetized precessional flows under weak stable stratification. Diffusive effects on the instability is considered in the simple case where the magnetic and thermal Prandtl numbers are both equal to one.

For smectic C* (SmC*) liquid crystals, configured in a bookshelf-type geometry between two horizontal parallel plates, with the bottom plate fixed and the top plate free to move, it is known from experiment that pumping can occur when an electric field is applied, i.e. an upward movement of the top plate through mechanical vibrations when the electric field is suddenly reversed. In this paper we revisit an earlier mathematical model for fast electric field reversal by removing an assumption made there on the velocity field; instead, we arrive at a time-dependent, two-dimensional squeeze-film model, which can ultimately be formulated in terms of a highly nonlinear integro-differential equation. Subsequent analysis leads to an unexpected solvability condition involving the five SmC* viscosity coefficients regarding the existence and uniqueness of solutions. Furthermore, we find that, when solutions do exist, they imply that the plate can move down as well as up, with the final resting position turning out to be dependent on the initial conditions; this is in stark contrast to the results of the earlier model.

Droplet vaporisation can exhibit distinct shrinkage kinetic laws depending on the experimental set-up and ambient conditions. In this work, we present a unified approach that combines experiment and theory to identify true shrinkage kinetics across a broad range of droplet vaporisation processes extending beyond the classical D2-law – particularly under realistic conditions involving support fibres or/and inevitable convective effects. Experimentally, we assume a power law $D^n= D_0^n- \textit{Kt}$, where K is the vaporisation rate constant, and re-express it as $(D/D_0)^n = 1 - t/t_{\textit{life}}$ in terms of the normalised droplet diameter $D/D_0$ and time t$ / $tlife relative to the droplet’s initial diameter D0 and lifetime tlife. Taking D as the diameter of a volume-equivalent sphere, the exponent n can be reliably extracted from the slope of the log–log plot of $( 1 - t/t_{\textit{life}})$ against $D/D_0$. The robustness of this method is demonstrated by re-confirming the D2-law for pure fuel droplet evaporation and validating the $D^{3/2} $-law for droplet evaporation under forced convection. We further apply this method to droplet combustion, revealing a significant departure from the D2-law with n $=$ 2.56 ± 0.20–2.65 ± 0.17 across various liquid fuels, unaffected by the presence of support fibres. An even more pronounced departure, with n approaching 3, is observed in droplet combustion within a continuous flame sustained by an auxiliary burner. Theoretically, we develop a more general theory to describe these droplet combustion processes, showing that the observed positive departures mainly result from flame-driven buoyant convection with 2.33 < n < 3, capturing well the experimental data. The same theoretical framework can also account for the negative departures in convection-driven vaporisation processes without flame, thereby providing a unified interpretation for the fundamental distinctions between flame-driven and non-flame-driven droplet vaporisation processes. The present study not only identifies distinct shrinkage power laws that emerge from complexities in these processes, but also reveals the central role of an inherent length scale – arising from underlying convective mechanisms – in shaping the true shrinkage kinetics that lead to violations of the D2-law.

A burning droplet in normal gravity inevitably encounters buoyant convection set up by the flame, which can significantly impact its shrinkage kinetics traditionally described by the D2-law. However, the detailed mechanism governing droplet vapourisation under such self-generated flame-driven buoyant convection remains elusive. Here, we present both experimental and theoretical evidence highlighting the critical role of buoyant convection in droplet combustion. Experimentally, we precisely measure the values of the shrinkage exponent n for various liquid fuels, revealing a significant departure from the D2-law. While the measured n values consistently fall within the narrow range 2.6–2.7, they exhibit a slight increase with the fuel’s boiling point. A more general and in-depth theory is also developed to explain such small but systematic variations, revealing that differences in flow and thermal boundary layer structures – arising from varying combustion intensities – may account for the observed trends. Our theory predicts three distinct values of n, namely 2.6, 8/3 ≈ 2.67 and 35/13 ≈ 2.69, successfully capturing slight differences in n among various fuels. This is the first study demonstrating that the shrinkage kinetics in droplet vapourisation driven by flame-induced buoyant convection is nearly universal, determined solely by the underlying transport mechanisms, although these can be significantly altered due to their high susceptibility to detailed fuel chemistry and combustion kinetics. The present theoretical framework not only enables accurate prediction and control of burning droplet behaviour, but also is extendable to analyse more complex combustion processes involving a broader range of fuel types and flow conditions.

Two-way diffusion equations arising in kinetic problems relating to electron scattering and in Brownian particle dynamics present singularities absent from conventional diffusion equations. Although calculations by Stein & Bernstein, and Fisch & Kruskal have revealed the formation of entry and exit slope discontinuities at the critical points where the velocity changes sign, the analytical structure of these discontinuities remains unclear. Here we fill this gap via a local similarity variable analysis, illustrated through the two-way diffusion equation $y \partial n/\partial x=\partial ^2 n/\partial y^2$ in $-1 \leq y \leq 1$; $0 \leq x \leq L$, with $n(x,\pm 1)=0$ with various entry conditions $n(0,y)_{y\gt 0}$, and the exit condition $n(L,y)_{y\lt 0}=0$. The similarity variable $\eta =y/x^{1/3}$ permits the analytical characterization of the entry discontinuity, except for constants determined by matching with numerical solutions obtained with two numerical schemes: separation of variables following the construction of Beals, or finite-difference discretization of the transient partial differential equation, which converges in time to a solution almost identical to the separation of variables solution. Although the slope discontinuity depends markedly on the initial condition $n(0,y)_{y\gt 0}$, a simple general similarity solution structure emerges empirically, always involving a spontaneous singular contribution $C |y|^{1/2}$ at $x=0,y\lt 0$. Slow convergence of both numerical solutions near $\{x,y\}=\{0,0\}$ is attributed to the poor eigenfunction representation of the ever-present singular solution component $|y|^{1/2}$. The similarity approach applies equally to other two-way diffusion equations when the coefficient of $\partial n/\partial x$ changes sign linearly with $y$. It can also be extended to situations where this coefficient is discontinuous at the critical points.

Surface roughness is often present in flight systems travelling at high speeds, but its interaction with compressible turbulence is not well understood. Using direct numerical simulations, we study how prism-shaped roughness influences supersonic turbulent boundary layers at a free-stream Mach number $M_\infty =2$. The dataset includes four simulations featuring cubic- and diamond-shaped elements in aligned and staggered configurations. All cases have an initial smooth region where a fully turbulent boundary layer transitions to a rough wall with positively skewed roughness elements relative to the smooth-wall zero plane. This causes a sudden boundary layer growth at the smooth-to-rough transition, generating an oblique shock wave. Individual roughness elements downstream do not generate shock or expansion waves, as they do not protrude into the supersonic region. For cubical elements, the staggered arrangement increases drag and produces more pronounced boundary layer growth than the aligned case. Rotating the cubes along their vertical axis further enhances these effects, yielding the highest drag. Interestingly, diamond-shaped elements in a staggered arrangement exhibit a dynamics similar to aligned cubes, producing lower drag than other cases. We explain the relative drag induced by each roughness shape by examining viscous and pressure drag components separately. The analysis reveals that, for staggered diamonds, the flow skims more easily over roughness, drastically reducing recirculation in troughs and gaps. In other cases, wake interactions are more prominent, causing spikes of highly positive and negative skin friction, a feature often neglected in reduced-order model formulations.

Particles in compressible shear flows experience lifting effects due to asymmetric pressure and viscous forces across the particle surface, rotation induced by asymmetric viscous forces (Magnus effect), and asymmetric compression and viscous effects if near a wall (wall effect). This work focuses on the lifting force on a solid spherical particle due to asymmetric pressure and shear stress distributions driven by density and velocity gradients. We show via direct numerical simulation and verify using scaling arguments that the lifting force in unbounded laminar compressible shear flows is a function of dynamic pressure gradient. We show that steady flow regimes demonstrate predictable lifting forces. Unsteady flow regimes demonstrate asymmetric vortex shedding which creates lift in directions not readily predictable. Thus, predicting lift requires the ability to predict wake structure. We develop approximate delineations between wake types at Reynolds numbers up to 20 000. We use the non-dimensional dynamic pressure gradient, Mach number, Reynolds number and predicted wake structure to develop a shear-induced lift model. The proposed model can be used in conjunction with a drag model to simulate particle motion in compressible shear flow.

For a perturbed trefoil vortex knot evolving under the Navier–Stokes equations, a sequence of $\nu$-independent times $t_m$ are identified that correspond to a set of scaled, volume-integrated vorticity moments $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}$, with this hierarchy $t_\infty \leqslant \ldots \leqslant t_m\ldots t_1=t_x\approx 40$ and $\mathcal{O}_{\textit{Vm}}=(\int _{V\ell }|\omega |^{2m}\,{\rm d}V)^{1/2m}$. For the volume-integrated enstrophy $Z(t)$, convergence of $\sqrt {\nu }Z(t)=\bigl (\nu ^{1/4}\mathcal{O}_{\textit{V}\text{1}}(t)\bigr )^2$ at $t_x=t_1$ marks the end of reconnection scaling. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. Meanwhile $Z$ accelerates, leading to approximate finite-time $\nu$-independent convergence of the energy dissipation rate $\epsilon (t)=\nu Z(t)$ at $t_\epsilon \sim 2t_x$. This is sustained over a finite temporal span of at least $\Delta T_\epsilon \searrow 0.5 t_\epsilon$, giving Reynolds number independent finite-time, temporally integrated dissipation, $\Delta E_\epsilon =\int _{\Delta T_\epsilon }\epsilon \,{\rm d}t$, and thus satisfies one definition for a dissipation anomaly, with enstrophy spectra that are consistent with transient $k^{1/3}$ Lundgren-like inertial scaling over some of the $\Delta T_\epsilon$ time. A critical factor in achieving these temporal convergences is how the computational domain $V_\ell =(2\ell \pi )^3$ is increased as $\ell \sim \nu ^{-1/4}$, for $\ell =2$ to 6, then to $\ell =12$, as $\nu$ decreases. Appendix A shows compatibility with established $(2\pi )^3$ mathematics where $\nu \equiv 0$ Euler solutions bound small $\nu$ Navier–Stokes solutions. Two spans of $\nu$ are considered. Over the first factor of 25 decrease in $\nu$, most of the $\nu ^{1/4}\mathcal{O}_{\textit{Vm}}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease (125 total) in $\nu$, with increased $\ell$ to $\ell =12$, there is initially only convergence of $\nu ^{1/4}\varOmega _{V\infty }(t)$ to $t_\infty$, without convergence for $9\gt m\gt 1$. Nonetheless, there is later $\sqrt {\nu }Z(t)$ convergence at $t_1=t_x$ and $\epsilon (t)=\nu Z$ over $t\sim t_\epsilon \approx 2t_x$.

Bayesian optimisation with Gaussian process regression was performed to optimise the shape of an elastically mounted cylinder undergoing transverse flow-induced vibration. The vibration amplitude and mean power coefficient were obtained from two-dimensional numerical simulations, with Reynolds number $Re = 100$. First, shape optimisation was performed to maximise the amplitude of undamped vibrations. The optimised shape was found to be a thin crescent cylinder aligned perpendicular to the oncoming flow. The optimised shapes exhibited simultaneous vortex-induced vibration and galloping, a response which was not observed for other cylinder geometries at the same Reynolds number. Shape optimisation was also performed to maximise the power coefficient, where the power generation device was modelled as a linear damper. The power-optimised cylinders were also thin crescents, but with greater curvature compared with the amplitude-optimised cylinders. Compared with a circular cylinder, improvements in the power coefficient and efficiency of up to $523\,\%$ and $152\,\%$, respectively, were obtained.

Unsteady aerodynamic forces in flapping wings arise from complex, nonlinear flow structures that challenge predictive modelling. In this work, we introduce a data-driven framework that links experimentally observed flow structures to sectional pressure loads on physical grounds. The methodology combines proper orthogonal decomposition and quadratic stochastic estimation (QSE) to model and interpret these forces using phase-resolved velocity fields from particle image velocimetry measurements. The velocity data are decomposed in a wing-fixed frame to isolate dominant flow features, and pressure fields are reconstructed by solving the Poisson equation for incompressible flows. The relationship between velocity and pressure modes is captured through QSE, which accounts for nonlinear interactions and higher-order dynamics. We introduce an uncertainty-based convergence criterion to ensure model robustness. Applied to a flapping airfoil, the method predicts normal and axial forces with less than 6 % average error using only two velocity modes. The resulting model reveals an interpretable underlying mechanism: linear terms in the QSE model the circulatory force linked to the formation of vortices on the wing, while quadratic terms capture the nonlinear component due to added-mass effects and flow–vorticity interactions. This data-driven yet physically grounded approach offers a compact tool for modelling the unsteady aerodynamics in flapping systems with potential to generalise to other problems.

We investigate how the addition of surfactant affects the governing equations for a bubble in a two-dimensional channel in the small-capillary-number limit. In the limit where the time scale for absorption of surfactant is much shorter than the time scales for transport of surfactant along the surface of the bubble, we derive a set of idealised free-surface boundary conditions that capture the effects of surfactant in a single dimensionless ‘elasticity parameter’, and apply them to the front and rear of the bubble separately. At the front of the bubble, there are three regions of interest: the front cap, the thin film region and a transition region that smoothly connects the other two regions. Through matched asymptotic expansions, we derive predictions for the thin film height and the pressure drop across the front meniscus. We find that the viscous pressure drop across the front meniscus is always larger for a surfactant-laden bubble than for a surfactant-free bubble, by an order-one factor of up to $4^{2/3}$. Using a similar analysis at the rear of the bubble, we find that the viscous pressure drop across the rear meniscus is also always larger in magnitude for a surfactant-laden bubble than for a surfactant-free bubble, again up to a maximum factor of $4^{2/3}$. Finally, we use these results to show that, for the same flow conditions, an isolated surfactant-laden bubble in a Hele-Shaw cell will travel more slowly than an isolated surfactant-free bubble.

Surface patterns on ablating materials are known to appear in both high-speed ground and flight tests, but the mechanisms behind their formation are not known. In this paper, the origins of surface patterns are investigated via a local linear stability analysis of compressible laminar boundary layers over a flat camphor plate. The effects of sublimation and conjugate heat transfer are included on both the baseflow and the linear fluctuations. This newly developed framework identifies a single mode that fully characterises the stability of the surface, and this surface mode becomes unstable under laminar conditions only when the wall temperature exceeds that of an adiabatic wall, $T_{\textit{ad}}$. These findings are consistent with experimental observations, where laminar flow conditions at adiabatic wall temperatures are observed to be stable. The present analysis also reveals that the nature of this surface mode varies as a function of the oblique angle $\psi = \tan ^{-1}({\beta /\alpha })$, where $\alpha$ and $\beta$ are the streamwise and spanwise wavenumbers. As the wall temperature increases, the most unstable orientation of the surface mode shifts from streamwise alignment ($\psi = 0$), towards the sonic angle ($\psi = \psi _s = \cos ^{-1}(1/M_e)$) and then towards spanwise alignment ($\psi = 90^\circ$). Finally, a critical wavenumber is identified (i.e. one at which the temporal growth rate reaches a maximum) which implies the formation of a surface pattern of a specific wavelength and orientation.

Heat-transfer measurements published in the literature seem to be contradictory, some showing a transition for the dependance of the Nusselt number (${\textit{Nu}}$) with the Rayleigh number (${\textit{Ra}}$) behaviour at ${\textit{Ra}} \approx 10^{11}$, some showing a delayed transition at higher ${\textit{Ra}}$, or no transition at all. The physical origin of this discrepancy remains elusive, but is hypothesised to be a signature of the multiple possible flow configurations for a given set of control parameters, as well as the sub-critical nature of the transition to the ultimate regime (Roche 2020 New J. Phys. vol. 22, 073056; Lohse & Shishkina 2023 Phys. Today vol. 76, no. 11, 26–32). New experimental and numerical heat-flux and velocity measurements, both reaching ${\textit{Ra}}$ up to $10^{12}$, are reported for a wide range of operating conditions, with either smooth boundaries, or mixed smooth–rough boundaries. Experiments are run in water at $40\,^\circ \textrm {C}$ (Prandtl number, ${\textit{Pr}}$ is 4.4), or fluorocarbon at $40\,^\circ \textrm {C}$ (${\textit{Pr}}$ is 12), and aspect ratios 1 or 2. Numerical simulations implement the Boussinesq equations in a closed rectangular cavity with a Prandtl number 4.4, close to the experimental set-up, also with smooth boundaries, or mixed smooth–rough boundaries. In the new measurements in the rough part of the cell, the Nusselt number is compatible with a ${\textit{Ra}}^{1/2}$ scaling (with logarithmic corrections), hinting at a purely inertial regime. In contrast to the ${\textit{Nu}}$ vs ${\textit{Ra}}$ relationship, we evidence that these seemingly different regimes can be reconciled: the heat flux, expressed as the flux Rayleigh number, ${\textit{Ra}}\textit{Nu}$, recovers a universal scaling with Reynolds number, which collapses all data, both our own and those in the literature, once a universal critical Reynolds number is exceeded. This universal collapse can be related to the universal dissipation anomaly, observed in many turbulent flows (Dubrulle 2019 J. Fluid Mech. vol. 867, no. P1, 1).

The spatial organisation of a passive scalar plume originating from a point source in a turbulent boundary layer is studied to understand its meandering characteristics. We focus shortly downstream of the isokinetic injection ($1.5\leqslant x/\delta \leqslant 3$, $\delta$ being the boundary-layer thickness) where the scalar concentration is highly intermittent, the plume rapidly meanders and breaks up into concentrated scalar pockets due to the action of turbulent structures. Two injection locations were considered: the centre of the logarithmic region and the wake region of the boundary layer. Simultaneous quantitative acetone planar laser-induced fluorescence and particle image velocimetry were performed in a wind tunnel, to measure scalar mixture fraction and velocity fields. Single- and multi-point statistics were compared with established works to validate the diagnostic novelties. Additionally, the spatial characteristics of plume intermittency were quantified using ‘blob’ size, shape, orientation and mean concentration. It was observed that straining, breakup and spatial reorganisation were the primary plume-evolution modes in this region, with little small-scale homogenisation. Further, the dominant role of coherent vortex motions in plume meandering and breakup was evident. Their action is found to be the primary mechanism by which the injected scalar is transported away from the wall in high concentrations (‘large meander events’). Strong spatial correlation was observed in both instantaneous and conditional fields between the high-concentration regions and individual vortex heads. This coherent transport was weaker for wake injection, where the plume only interacts with outer vortex motions. A coherent-structure-based mechanism is suggested to explain these transport mechanisms.

Hydrodynamic instability can occur when a viscous fluid is driven rapidly through a flexible-walled channel, including a multiplicity of steady states and distinct families of self-excited oscillations. In this study we use a computational method to predict the stability of flow through a planar finite-length rigid channel with a segment of one wall replaced by a thin pre-tensioned elastic beam of negligible mass. For large external pressures, this system exhibits a collapsed steady state that is unstable to low-frequency self-excited oscillations, where the criticality conditions are well approximated by a long-wavelength one-dimensional (1-D) model. This oscillation growing from a collapsed state exhibits a reduced inlet driving pressure compared with the corresponding steady flow, so the oscillating state is energetically more favourable. In some parameter regimes this collapsed steady state is also unstable to distinct high-frequency normal modes, again predicted by the 1-D model. Conversely, for lower external pressures, the system exhibits an inflated steady state that is unstable to another two modes of self-excited oscillation, neither of which are predicted by the lower-order model. One of these modes becomes unstable close to the transition between the upper and lower steady states, while the other involves small-amplitude oscillations about a highly inflated wall profile with large recirculation vortices within the cavity. These oscillatory modes growing from an inflated steady state exhibit a net increase in driving pressure compared with the steady flow, suggesting a different mechanism of instability to those growing from a collapsed state.

This article follows on from Scott & Cambon (J. Fluid Mech., vol. 979, 2024, A17) and Scott (Phys. Rev. E, vol. 111, 2025, 035101). Like those articles, it concerns weak, decaying homogeneous turbulence in a rotating, stably stratified fluid with constant Brunt–Väisälä frequency, $N$. The difference is that here we consider the case in which $\beta =2{\varOmega} /N$ is close to $1$, where ${\varOmega}$ is the rotation rate. Because this renders inertial-gravity waves only weakly dispersive, wave-turbulence theory, which played a prominent role in the earlier studies, no longer applies. Indeed, wave-turbulence analysis does not appear here. Nonetheless, much of the analytical framework, based on modal decomposition, carries over, as do many of the conclusions. The flow is expressed as a sum of wave and non-propagating (NP) modes and their weak-turbulence mode-amplitude evolution equations are derived for small $\beta -1$. The NP component is found to evolve independently of the wave one, following an amplitude equation which is precisely that of the previous studies in the limit $\beta \rightarrow 1$. The NP component induces coupling between wave modes and, without it, the wave component has purely linear decay. The mode-amplitude equations are integrated numerically using a scheme similar to that of classical direct numerical simulation and results given. We find an inverse energy cascade of the NP component, whereas the presence of that component induces a forward cascade, hence significant dissipation, of the wave component. Detailed results are given for the energy, energy spectra and energy fluxes of the two components.

The effectiveness of polymer drag reduction by targeted injection is studied in comparison with that of a uniform concentration (or polymer ocean) in a turbulent channel flow. Direct numerical simulations are performed using a pseudo-spectral code to solve the coupled equations of a viscoelastic fluid using the finitely extensible nonlinear elastic dumbbell model with the Peterlin approximation. Light and heavy particles are used to carry the polymer in some cases, and polymer is selectively injected into specific flow regions in the other cases. Drag reduction is computed for a polymer ocean at a viscosity ratio of $\beta = 0.9$ for simulation validation, and then various methods of polymer addition at $\beta = 0.95$ are compared for their drag-reduction performance and general effect on the flow. It was found that injecting polymer directly into regions of high axial strain inside and around coherent vortical structures was the most effective at reducing drag, while injecting polymer very close to the walls was the least effective. The targeting methods achieved up to 2.5 % higher drag reduction than an equivalent polymer ocean, offering a moderate performance boost in the low drag-reduction regime.

Shock trains compress incoming supersonic flow through a series of shock wave/turbulent boundary layer interactions (STBLIs) that occur in rapid streamwise succession. In this work, the global flow changes across the entire turbulent shock train are analysed as the confluence of local changes imparted by individual STBLIs. For this purpose, wall-resolved large eddy simulations are used on a constant area, back-pressured channel configuration with an entry Mach number of 2.0. Local changes due to individual STBLIs are evaluated in terms of deviations from incoming, near-equilibrium boundary layers, by systematically examining properties of the mean flow structure and turbulent statistics. The first STBLI in the train induces a strongly separated region, which interrupts inner layer dynamics and incites wall-normal deflection of turbulent structures, leading to prominent outer layer Reynolds stress amplification and related transport phenomena. Downstream of the first STBLI, the thickened, turbulent wall layer repeatedly interacts with subsequent shock waves in the train, resulting in cyclic attenuation and amplification of turbulent stresses, localised incipient separations and variations in mean momentum flux gradients. Decreases in mean Mach number along the shock train result in downstream shocks weakening to the point that interactions with the turbulent boundary layer impart negligible changes on the local flow. Consequently, after a sufficient streamwise extent, the boundary layers asymptote towards a new equilibrium state, thus recovering certain classical properties of near-wall turbulence. Among the features that reappear are a self-similar, adverse pressure gradient velocity profile and the restoration of the autonomous roller-streak cycle.