This study explores precession-driven flows in a non-axisymmetric ellipsoid spinning around its medium axis. Using an experimental approach, we focus on two aspects of the flow: the base flow of uniform vorticity and the development of fluid instabilities. In contrast to a preceding paper (J. Fluid. Mech., vol. 932, 2022, A24), where the ellipsoid rotated around its shortest axis, we do not observe bi-stability or hysteresis of the base flow, but a continuous transition from small to large differential rotation and tilt of the fluid rotation axis. We then use the model developed by Noir & Cébron (J. Fluid. Mech., vol. 737, 2013, pp. 412–439) to numerically determine regions in the parameter space of axial and equatorial deformations for which bi-stability may exist. Concerning fluid instabilities, we use three independent observations to track the onset of both boundary layer and parametric instabilities. Our results clearly show the presence of a parametric instability, yet the exact nature of the underlying mechanism (conical shear layer instability, shear instability and elliptical instability) is not unambiguously identified. A coexisting boundary layer instability, although unlikely, cannot be ruled out based on our experimental data. To make further progress on this topic, a new generation of experiments at significantly lower Ekman numbers (ratio of rotation and viscous time scales) is clearly needed.

A novel selective mode decomposition, proper orthogonal decomposition and dynamic mode decomposition methods are used to analyse large-eddy simulation data of the flow field about a NACA0012 airfoil at low Reynolds numbers of $5\times 10^4$ and $9\times 10^4$, and at near-stall conditions. The objective of the analysis is to investigate the structure of the laminar separation bubble (LSB) and its associated low-frequency flow oscillation (LFO). It is shown that the flow field can be decomposed into three dominant flow modes: two low-frequency modes (LFO-Mode-1 and LFO-Mode-2) that govern an interplay of a triad of vortices and sustain the LFO phenomenon, and a high-frequency oscillating (HFO) mode featuring travelling Kelvin–Helmholtz waves along the wake of the airfoil. The structure and dynamics of the LSB depend on the energy content of these three dominant flow modes. At angles of attack lower than the stall angle of attack and above the angle of a full stall, the flow is dominated by the HFO mode. At angles of attack above the stall angle of attack the LFO-Mode-2 overtakes the HFO mode, triggers instability in the LSB and initiates the LFO phenomenon. Previous studies peg the structure, stability and bursting conditions of the separation bubble to local flow parameters. However, the amplitude of these local flow parameters is dependent on the energy content of the three dominant flow modes. Thus, the present work proposes a more robust bursting criterion that is based on global eigenmodes.

We perform a detailed numerical study of modal and non-modal stability in oblique Couette–Poiseuille profiles, which are among the simplest examples of three-dimensional boundary layers. Through a comparison with the Orr–Sommerfeld operator for the aligned case, we show how an effective wall speed succinctly characterizes modal stability. Large-scale parameter sweeps reveal that the misalignment between the pressure gradient and wall motion is, in general, destabilizing. For flows that are sufficiently oblique, the instability is found to depend exclusively on the direction of wall motion and not on its speed, a conclusion supported, in part, by the perturbation energy budget and the evolution of the critical layers. Closed forms for the critical parameters in this regime are derived using a simple analysis. From a non-modal perspective, pseudoresonance is examined through the resolvent and the $\epsilon$-pseudospectra. An analysis of the unforced initial value problem shows that the maximum energy gain is highly dependent on both the magnitude and direction of the wall velocity. However, the strongest amplification is always achieved for configurations that are only weakly skewed. Finally, the optimal perturbations appear to develop via a lift-up effect enhanced by an Orr-like mechanism, the latter driven by cross-flow shear.

We investigated the effects of two-dimensional sharp-edged rectangular bumps on Tollmien–Schlichting (TS) wave evolution using direct numerical simulation. The bump height, $h$, ranged from 5 % to 40 % of the local displacement thickness, $\delta ^*$. Behind the bump, a recirculating flow region could be formed whose length increased nonlinearly with $h$. The bump height effect on the TS wave, which was the dominant, scaled super-exponentially with $h$. We also showed a substantial effect of the $\delta ^*$-based Reynolds number, ${\textit {Re}} _{\delta ^*}$. Firstly, the bump wake extended with ${\textit {Re}} _{\delta ^*}$, promoting larger TS wave growth rates. The second effect is related to proximity to the upper branch of the instability loop, accounting for the influence of the TS frequency, as well. It dictates the bump impact increases as it gets closer to transition, either by the bump moving downstream or the transition moving upstream. For a 40 % high bump, for example, changing the ${\textit {Re}} _{\delta ^*}$ at the bump location from 1500 to 2000 increased $\Delta N$ by a factor of 2 ($\Delta N$ represents a measure of a surface irregularity effect on the smooth plate N-factor). We also found that $(\Delta N)_{max}$ increases linearly with ${\textit {Re}} _{hh}$. Results in the subsonic regime showed that the bump impact attenuates with Mach number up to 0.7 but above it, stabilisation is surpassed by the destabilising effect caused by the recirculation lengthening. This is mostly associated with the bump wake that extends with the pressure gradient which increases substantially towards the sonic speed. This is enhanced if the surface is adiabatic rather than isothermal.

The free-stream turbulence (FST) induced transition in perfect and non-ideal gas zero-pressure-gradient flat-plate boundary layers is investigated by means of large-eddy simulations. The study focuses on the influence of large incoming disturbances over the laminar-to-turbulent transition, by comparing two different integral length scales $L_f$, which differ by a factor of seven, at different FST intensities $T_u$. High-subsonic dense-gas boundary layers of the organic vapour Novec649, representative of organic Rankine cycle applications, are compared with air flows at Mach numbers 0.1 and 0.9. Compressibility and non-ideal gas effects are shown to be of minor importance in comparison to the influence of the FST integral length scale $L_f$. An increase of the inlet turbulent intensity always promotes transition, whereas an increase of $L_f$ has a double effect on the transition onset. At $T_u=2.5\,\%$, increasing $L_f$ promotes the transition, while it tends to delay transition for an FST intensity of 4 %. Larger FST integral scales tend to increase the spanwise distance between laminar streaks generated in the boundary layer. Two competing transition scenarios are observed. When the incoming turbulence intensity and length scale are moderate, the classical bypass route consists in the linear non-modal growth of streaks, which then experience secondary instabilities (sinuous or varicose) and lead to the generation of turbulent spots. The second scenario is characterized by the appearance of $\Lambda$-shaped structures near the inlet, which are further stretched to hairpin vortices before breaking down to turbulence. Spot inceptions can therefore occur at earlier locations than the streak growth. We are then faced with a competition between the classical bypass transition and nonlinear response mechanisms that ‘bypass’ this route. The present case at high $L_f$ and low $T_u$ is an example of a competing scenario, but even for the higher $T_u$ and $L_f$ conditions, only approximately one-third of turbulent spots are due to the $\Lambda$-shaped events. The nonlinear alternative route has strong similarities with scenarios described previously in the literature in the presence of leading edge effects or due to passing wakes. Such a path is governed by the turbulence intensity, but also by the integral length scale, with both parameters playing a critical role in the generation of the $\Lambda$-shaped structures near the inlet. This alternative mechanism is found to be robust under varying flow and thermodynamic conditions.

It is well known that buoyancy suppresses, and can even laminarise, turbulence in upward heated pipe flow. Heat transfer seriously deteriorates in this case. A new direct numerical simulation model is established to simulate flow-dependent heat transfer in an upward heated pipe. The model shows good agreement with experimental results. Three flow states are simulated for different values of the buoyancy parameter $C$: shear turbulence, laminarisation and convective turbulence. The latter two regimes correspond to the heat transfer deterioration regime and the heat transfer recovery regime, respectively (Jackson & Li 2002; Bae et al., Phys. Fluids, vol. 17, issue 10, 2005; Zhang et al., Appl. Energy, vol. 269, 2020, 114962). We confirm that convective turbulence is driven by a linear instability (Su & Chung, J. Fluid Mech., vol. 422, 2000, pp. 141–166) and that the deteriorated heat transfer within convective turbulence is related to a lack of rolls near the wall, which leads to weak mixing between the flow near the wall and the centre of the pipe. Having surveyed the fundamental properties of the system, we perform a nonlinear non-modal stability analysis, which seeks the minimal perturbation that triggers a transition from the laminar state. Given the differences between shear and convective turbulence, we aim to determine how the nonlinear optimal (NLOP) changes as the buoyancy parameter $C$ increases. We find that at first, the NLOP becomes thinner and closer to the wall. Most importantly, the critical initial energy $E_0$ required to trigger turbulence keeps increasing, implying that attempts to trigger it artificially may not be an efficient means to improve heat transfer at larger $C$. At $C=6$, a new type of NLOP is discovered, capable of triggering convective turbulence from lower energy, but over a longer time. It is active only in the centre of the pipe. We next compare the transition processes, from linear instability and by the nonlinear non-modal excitation. At $C=4$, linear instability leads to a state that approaches a travelling wave solution or periodic solutions, while the minimal seed triggers shear turbulence before decaying to convective turbulence. Deeper into the parameter space for convective turbulence, at $C=6$, the new nonlinear optimal triggers convective turbulence directly. Detailed analysis of the periodic solution at $C=4$ reveals three stages: growth of the unstable eigenfunction, the formation of streaks, and the decay of the streaks. The stages of the cycle correspond to changes in the linear instability of the turbulent mean velocity profile. Unlike the self-sustaining process for classical shear flows, where the streak is disrupted via instability, here, decay of the streak is more closely linked to suppression of the linear instability of the mean flow, and hence suppression of the rolls. Flow visualisations at $C$ up to $10$ also show similar processes, suggesting that the convective turbulence in the heat transfer recovery regime is sustained by these three typical processes.

We investigate the dynamics of a columnar Taylor–Green vortex array under strong stratification, focusing on Froude numbers $0.125\leq Fr \leq 1.0$, with the aim of identifying and understanding the primary instabilities that lead to the vortices’ breakdown. Linear stability analysis reveals that the fastest-growing vertical wavenumber scales with $Fr^{-1}$, while the dimensionless growth rate remains approximately constant. The most unstable eigenmode, identified as the mixed hyperbolic mode by Hattori et al. (J. Fluid Mech., vol. 909, 2021, A4), bears significant similarities to the zigzag instability, first discovered by Billant & Chomaz (J. Fluid Mech., vol. 418, 2000, pp. 167–188). Direct numerical simulations further confirm that the zigzag instability is crucial in amplifying initial random perturbations to finite amplitude, with the flow structure and modal growth rate consistent with the linear stability analysis. In particular, the characteristic vertical length scale of turbulence matches that of the fastest-growing linear mode. These findings underscore the broader relevance of the zigzag instability mechanism beyond its initial discovery in vortex pairs, demonstrating its role in facilitating direct energy transfer from vertically uniform vortical motions to a characteristic vertical length scale proportional to $Fr$ in strongly stratified flows.

Linear stability analysis currently fails to predict turbulence transition in canonical viscous flows. We show that two alternative models of the boundary condition for incipient perturbations at solid walls produce linear instabilities that could be sufficient to explain turbulence transition. In many cases, the near-wall behaviour of the discovered instabilities is empirically indistinguishable from the classical no-slip condition. The ability of these alternative boundary conditions to predict linear instabilities that are consistent with turbulence transition suggests that the no-slip condition may be an overly simplified model of fluid–solid interface physics, particularly as a description of the flow perturbations that lead to turbulence transition in wall-bounded flows.

Numerous studies have indicated that turbulence typically initiates along the boundary layer of the stationary disk within a rotor–stator cavity. To describe the transition process to turbulence on the stationary side of a closed rotor–stator cavity, a comprehensive approach combining global linear stability analysis with direct numerical simulation was adopted in the present study. The proposed model aligns with that of Yim et al. (J. Fluid Mech., vol. 848, 2018, pp. 631–647), who investigated the stability characteristics of the rotating-disk boundary layer in a rotor–stator cavity. In order to achieve a stable inflow for the stationary-disk boundary layer, we rotate the shroud together with the rotating disk. Through careful global stability analysis, the predominant spiral mode exhibiting the highest instability in the boundary layer of the stationary disk was discerned, corroborating observations from simulations. Initially, the spiral mode undergoes linear amplification, reaches a state of linear saturation and enters the nonlinear regime. Following nonlinear saturation in the flow field, a circular wave mode arises due to the influence of mean flow distortion. As the Reynolds number attained a sufficiently high level, the interplay between the downstream-propagating circular mode and spiral mode amplified disturbances in the boundary layer of the stationary disk, ultimately leading to the development of localised turbulence at the mid-radius of the rotor–stator cavity. Notably, the present study is the first to elucidate the coexistence of laminar–transitional–turbulent flow states in the stationary-disk boundary layer through direct numerical simulations.

Direct numerical simulations (DNS) were carried out to investigate flow control for transition delay using steady blowing/suction strips at the wall of a flared cone at Mach 6 and zero angle of attack. For the numerical investigations of the transition control strategy, the flared cone geometry and the flow conditions of the experiments in the Boeing/Air Force Office of Scientific Research (AFOSR) Mach 6 Quiet Tunnel (BAM6QT) at Purdue University were chosen. For the DNS, transition was initiated by introducing random disturbances at the inflow of the computational domain, emulating ‘natural’ transition in wind-tunnel experiments caused by free-stream noise. In both wind-tunnel experiments and numerical simulations, streamwise ‘hot’ streaks were found on the surface of the flared cone, which are caused by a nonlinear interaction of an axisymmetric second-mode wave and a pair of oblique waves of the same frequency (‘fundamental resonance’). The objective of the flow control strategy proposed here is to delay the transition onset, and thus mitigate the negative consequences associated with the nonlinear transition stages, i.e. the development of hot streaks and large wall-pressure amplitudes that were observed in experiments and DNS. Our previous so-called ‘controlled’ transition simulations have shown that flow control using steady blowing and suction strips can lead to a significant delay of the hot streak development on the surface of the flared cone. The simulation results presented in this paper show that this flow control strategy remains effective, even in a natural transition scenario characterized by broadband disturbances.

Numerical simulations of thermoelectrohydrodynamic convection in a dielectric liquid inside a finite-length cylindrical annulus with a fixed temperature difference have been performed with increasing high-frequency electric tension under microgravity conditions. The electric field, coupled with the permittivity gradient, generates a dielectrophoretic buoyancy force whose non-conservative part can induce thermoelectric convection in the liquid. The liquid remains in a conductive state below a critical value of the applied electric voltage. At a critical value, a supercritical bifurcation occurs from the conductive state to a convective state made of stationary helicoidal vortices. A further increase of electric voltage leads to oscillatory helicoidal vortices and then to wavy patterns before spoke patterns dominate the convective flow. The dielectrophoretic force is shown to enhance the heat transfer from the hot to cold walls due to induced convective flows. Particularly, these results demonstrate that the dielectrophoretic buoyancy force holds promise to replace the gravitational force to induce efficient heat transfer in microgravity conditions, and they contribute to a better fundamental understanding of heat transfer in microgravity.

The efficacy of steady large-amplitude blowing/suction on instability and transition control for a hypersonic flat plate boundary layer with Mach number 5.86 is investigated systematically. The influence of the blowing/suction flux and amplitude on instability is examined through direct numerical simulation and resolvent analysis. When a relatively small flux is used, the two-dimensional instability critical frequency that distinguishes the promotion/suppression mode effect closely aligns with the synchronisation frequency. For the oblique wave, as the spanwise wavenumber increases, the suppression effects would become weaker and the mode suppression bandwidth diminishes/increases in general in the blowing/suction control. Increasing the blowing/suction flux can effectively broaden the frequency bandwidth of disturbance suppression. The influence of amplitude on disturbance suppression is weak in a scenario of constant flux. To gain a deeper insight into disturbance suppression mechanism, momentum potential theory (MPT) and kinetic energy budget analysis are further employed in analysing disturbance evolution with and without control. When the disturbance is suppressed, the blowing induces the transport of certain acoustic components along the compression wave out of the boundary layer, whereas the suction does not. The velocity fluctuations are derived from the momentum fluctuations of the MPT. Compared with the momentum fluctuations, the evolutions indicated by each component's velocity fluctuations greatly facilitate the investigations of the acoustic nature of the second mode. The rapid variation of disturbance amplitude near the blowing is caused by the oscillations of the acoustic component and phase speed differences between vortical and thermal components. Kinetic energy budget analysis is performed to address the non-parallel effect of the boundary layer introduced by blowing/suction, which tends to suppress disturbances near the blowing. Moreover, viscous effects leading to energy dissipation are identified to be stronger in regions where the boundary layer is rapidly thickening. Finally, it is demonstrated that a flat plate boundary layer transition triggered by a random disturbance can be delayed by a blowing/suction combination control. The resolvent analysis further demonstrates that disturbances with frequencies that dominate the early transition stage are dampened in the controlled base flow.

We study the stability of a zero-pressure gradient boundary layer subjected to free-stream disturbances by means of local stability analysis. The dataset under study corresponds to a direct numerical simulation (DNS) of a flat plate with a sharp leading edge in realistic wind tunnel conditions, with a turbulence level of 3.45 % at the leading edge. We present a method to track the convective evolution of the secondary instabilities of streaks by performing sequential stability calculations following the wave packet, connecting successive unstable eigenfunctions. A scattered nature, in time and space, of secondary instabilities is seen in the stability calculations. These instabilities can be detected before they reach finite amplitude in the DNS, preceding the nucleation of turbulent spots, and whose appearance is well correlated to the transition onset. This represents further evidence regarding the relevance of secondary instabilities of streaks in the bypass transition in realistic flow conditions. Consistent with the spatio-temporal nature of this problem, our approach allows us to integrate directly the local growth rates to obtain the spatial amplification ratio of the individual instabilities, where it is shown that instabilities reaching an $N$-factor in the range [2.5,4] can be directly correlated to more than 65 % of the nucleation events. Interestingly, it is found that high amplification is not only attained by modes with high growth rates, but also by instabilities with sustained low growth rates for a long time.

We investigate the global instability mechanism of the flow past a three-dimensional stepped cylinder. A comprehensive study is performed for different diameter ratios of the two joined cylinders ($r=D/d$) ranging from $r=1.1$ to $r=4$. Independently of $r$, the spectrum of the linearised Navier–Stokes operator reveals a pair of complex conjugate eigenvalues, with Strouhal number $St \approx 0.11$. The initial transition is triggered by a two-dimensional mechanism of the larger cylinder only, not affected by the presence of the junction and the smaller cylinder ($Re_{D,cr}\approx 47$). The structural sensitivity analysis is used to identify where the instability mechanism acts. The onset of transition is solely localised in the large cylinder wake (L cell), where the wavemaker has two symmetric lobes across the separation bubble. When the Reynolds number increases, a second and a third unstable pair of complex conjugate eigenvalues appears. They are localised in the small cylinder (S) wake and modulation (N) region. For any $r$, the appearance of unstable eigenmodes resembling the three cells S–N–L in the wake is observed. The nonlinear simulation results support this finding, in contrast with the previous classification of the laminar vortex shedding in direct (L–S) and indirect (L–N–S) modes interaction Lewis & Gharib (Phys. Fluids, vol. 4, 1992, pp. 104–117). This result indicates that each cell undergoes a supercritical Hopf bifurcation for any $r$. As $r$ approaches $1$, the modal linear stability results also show an unstable eigenmode in the wake of the small cylinder resembling a new modulation cell, named N2, similar to the N cell but mirrored with respect to the junction plane.

This paper reports on the efficacy of the Görtler number in scaling the laminar-turbulent boundary-layer transition on rotating cones facing axial inflow. Depending on the half-cone angle $\psi$ and axial flow strength, the competing centrifugal and cross-flow instabilities dominate the transition. Traditionally, the flow is evaluated by using two parameters: the local meridional Reynolds number $Re_l$ comparing the inertial versus viscous effects and the local rotational speed ratio $S$ accounting for the boundary-layer skew. We focus on the centrifugal effects, and evaluate the flow fields and reported transition points using Görtler number based on the azimuthal momentum thickness of the similarity solution and local cone radius. The results show that Görtler number alone dominates the late stages of transition (maximum amplification and turbulence onset phases) for a wide range of investigated $S$ and half-cone angle ($15^{\circ } \leq \psi \leq 50^{\circ }$), although the early stage (critical phase) seems to be not determined by the Görtler number alone on the broader cones ($\psi =30^{\circ }$ and $50^{\circ }$) where the primary cross-flow instability dominates the flow. Overall, this indicates that the centrifugal effects play an important role in the boundary-layer transition on rotating cones in axial inflow.

We investigate the influence of vortices remote from the boundary on the near-wall flow dynamics in wall-bounded flows. A vortex ring with precisely controlled local twist is introduced into the outer layer of a channel flow at a moderate Reynolds number. We find that the minimum vorticity flux for triggering the transition to turbulence is significantly reduced from the initial disturbance of an untwisted vortex ring to that of a twisted ring. In particular, the latter disturbance can cause vortex bursting in the early transitional stage. The impact of vortex bursting on the transition process is characterised by the near-wall, wall-normal velocity with the rapid distortion theory. The wall-normal velocity grows during vortex bursting, and leads to streak formation and then the transition to turbulence. The notable wall-normal velocity is induced by the large di-vorticity generated in vortex bursting. We model the growing radial component of the di-vorticity in terms of the local twist, and demonstrate that its surge is due to the generation of highly twisted vortex lines in vortex bursting. Then, we derive that the generation of the di-vorticity in the outer layer enhances the wall-normal velocity in the inner layer via the Poisson equation with the image method and the multipole expansion. Thus, we elucidate that the vortex bursting can have an effect on the transition process.

This paper investigates the linear and nonlinear evolution of radiating modes under the influence of the spontaneously emitted Mach waves in a simple set-up of the supersonic boundary layers that develop in the entry region of a channel formed by two parallel semi-infinite flat plates. Two scenarios are considered. The first occurs in the boundary layers having identical wall conditions, where the Mach wave emitted by a radiating mode in one boundary layer influences the instability in the other. The second scenario takes place when the wall temperatures are different, in which case the spontaneously radiated Mach wave is reflected by the other boundary layer back to act on the radiating mode. Appropriate amplitude equations with the acoustic feedback effect being accounted for are derived. In each case, the effect of the spontaneously emitted sound contributes a linear term of delay type to the respective amplitude equation. For the first scenario, analytical and numerical studies of the amplitude equations show that due to the back action of the spontaneously radiated Mach wave, the amplitude exhibits rapid oscillations, and in the case of enhanced feedback effects, its envelope experiences near extinction followed by resurrection. The study of the coupled equations shows that the two modes with different initial amplitudes either undergo oscillations before attenuating, or terminate a finite-distance singularity at different locations. For the second scenario, the acoustic feedback produces similar effects in a broad range of wall temperature. The effects become pronounced, and the dependence on the wall temperature becomes more sensitive when the latter approaches the value corresponding to the resonance. Estimates suggest that such acoustic feedback is likely to be present in typical wind tunnel experiments and models for scramjet combustors.

An experimental study in a wind tunnel is presented to explore the wake of a floating wind turbine subjected to harmonic side-to-side and fore–aft motions under laminar inflow conditions. The wake recovery is analysed as a function of the frequency of motion $f_p$, expressed by the rotor-based Strouhal number, $St = f_p D / U_{\infty }$ ($D$ is the rotor diameter, $U_{\infty }$ the inflow wind speed). Our findings indicate that both directions of motion accelerate the transition to the far-wake compared with the fixed turbine. The experimental outcomes confirm the computational fluid dynamics results of Li et al. (J. Fluid Mech., vol. 934, 2022, p. A29) showing that sideways motions lead to faster wake recovery, especially for $St \in [0.2, 0.6]$. Additionally, we find that fore–aft motions also lead to better recovery for $St \in [0.3, 0.9]$. The recovery is closely linked to nonlinear spatiotemporal dynamics found in the shear layer region of the wake. For both directions of motion and $St \in [0.2, 0.55]$, the noisy wake dynamics lock in to the frequency of the motion. In this synchronised-like state, sideways motions result in large coherent structures of meandering, and fore–aft movements induce coherent pulsing of the wake. For fore–aft motion and $St \in [0.55, 0.9]$, the wake shows a more complex quasiperiodic dynamic, namely, a self-generated meandering mode emerges, which interacts nonlinearly with the excitation frequency $St$, as evidenced by the occurrence of mixing components. The coherent structures grow nonlinearly, enhance wake mixing and accelerate the transition to the far-wake, which, once reached, exhibits universal behaviour.

We investigate the linear instability of two-layer stratified shear flows in a sloping two-dimensional channel, subject to non-zero longitudinal gravitational forces. We reveal three previously unknown instabilities, distinct from the well-known Kelvin–Helmholtz instability and Holmboe wave instability, in that they have longer wavelengths (of the order of 10 to $10^3$ shear-layer depths) and often slower growth rates. Importantly, they can grow in background flows with gradient Richardson number $\gg 1$, which offers a new mechanism to sustain turbulence and mixing in strongly stratified flows. These instabilities are shown to be generic and relatively insensitive to Reynolds number, Prandtl number, base flow profile and boundary conditions. The nonlinear evolution of these instabilities is investigated through a forced direct numerical simulation, in which the background momentum and density are sustained. The growth of long unstable waves in background flows initially stable to short wave causes a decrease in the local gradient Richardson number. This leads to local nonlinear processes that result in small-scale overturns resembling Kelvin–Helmholtz billows. Our results establish a new energy exchange pathway, where the mean kinetic energy of a strongly stratified flow is extracted by primary unstable long waves and secondary short waves, and subsequently dissipated into internal energy.

Floquet stability analysis and direct simulations of a circular cylinder undergoing vortex-induced vibration (VIV) are presented. Simulation predictions are examined for the reduced velocity range over which there is a strong and periodic resonant response: $U_r \in [4.0, 8.0]$, focusing on a mass ratio of $m^* = 2.546$ matching a number of previous investigations. Over most of this range, the dominant wake modes present are analogous to modes A, B and QP (quasi-periodic) observed in a stationary circular cylinder wake. However, at $U_r = 4.5$, the dominant modes are B, QP and a subharmonic mode (SH), whereas at $U_r = 4.0$, the two-dimensional base state switches to a $P+S$ wake. The critical Reynolds number for two- to three-dimensional transition is observed to decrease with an increase of $U_r$, in line with a decreasing response amplitude. Over this range, the minimum ${\textit {Re}}$ for which the wake remains two-dimensional is 202, which occurs at $U_r = 7.5$, but this increases to ${\textit {Re}}_{cr} = 300$ at $U_r = 4.5$, noting the critical Reynolds number for a stationary circular cylinder is ${\textit {Re}}_{cr}=189$. The corresponding critical spanwise wavelengths for $U_r = 4.5$ and 8 are $1.4D$ (mode B) and $4.0D$ (mode A), respectively. Simulations indicate that even at $Re=300$, flow three-dimensionality increases the amplitude of the lower branch considerably. The investigation establishes the role of oscillation amplitude and reduced velocity in three-dimensional transition for elastically mounted systems.