Generalised two-dimensional (2-D) fluid dynamics is characterised by a relationship between a scalar field $q$, called generalised vorticity, and the stream function $\psi$,namely $q = (-\nabla ^2)^{\frac {\alpha }{2}} \psi$. We study the transition of cascades in generalised 2-D turbulence by systematically varying the parameter $\alpha$ and investigating its influential role in determining the directionality (inverse, forward or bidirectional) of these cascades. We derive upper bounds for the dimensionless dissipation rates of generalised energy $E_G$ and enstrophy $\Omega _G$ as the Reynolds number tends to infinity. These findings corroborate numerical simulations, illustrating the inverse cascade of $E_G$ and forward cascade of $\Omega _G$ for $\alpha \gt 0$, contrasting with the reverse behaviour for $\alpha \lt 0$. The dependence of dissipation rates on system parameters reinforces these observed transitions, substantiated by spectral fluxes and energy spectra, which hint at Kolmogorov-like scalings at large scales but discrepancies at smaller scales between numerical and theoretical estimates. These discrepancies are possibly due to non-local transfers, which dominate the dynamics as we go from positive to negative values of $\alpha$. Intriguingly, the forward cascade of $E_G$ for $\alpha \lt 0$ reveals similarities to three-dimensional turbulence, notably the emergence of vortex filaments within a 2-D framework, marking a unique feature of this generalised model.

Rogue waves (RWs) can form on the ocean surface due to the well-known quasi-four-wave resonant interaction or superposition principle. The first is known as the nonlinear focusing mechanism and leads to an increased probability of RWs when unidirectionality and narrowband energy of the wave field are satisfied. This work delves into the dynamics of extreme wave focusing in crossing seas, revealing a distinct type of nonlinear RWs, characterised by a decisive longevity compared with those generated by the dispersive focusing (superposition) mechanism. In fact, through fully nonlinear hydrodynamic numerical simulations, we show that the interactions between two crossing unidirectional wave beams can trigger fully localised and robust development of RWs. These coherent structures, characterised by a typical spectral broadening then spreading in the form of dual bimodality and recurrent wave group focusing, not only defy the weakening expectation of quasi-four-wave resonant interaction in directionally spreading wave fields, but also differ from classical focusing mechanisms already mentioned. This has been determined following a rigorous lifespan-based statistical analysis of extreme wave events in our fully nonlinear simulations. Utilising the coupled nonlinear Schrödinger framework, we also show that such intrinsic focusing dynamics can be captured by weakly nonlinear wave evolution equations. This opens new research avenues for further explorations of these complex and intriguing wave phenomena in hydrodynamics as well as other nonlinear and dispersive multi-wave systems.

The dependence of the Richtmyer–Meshkov instability (RMI) on post-shock Atwood number ($A_1$) is experimentally investigated for a heavy–light single-mode interface. We create initial interfaces with density ratios of heavy to light gases ranging from 1.73 to 34.07, and achieve the highest $|A_1|$ value reported to date for gaseous-interface experiments (0.95). For the first time, spike acceleration is observed in experiments with a heavy–light configuration. The models for the start-up, linear and weakly nonlinear evolution stages are evaluated over a wide range of $A_1$ conditions. Specifically, the models proposed by Li et al. (Phys. Fluids, vol. 36, 2024, 056104) and Wouchuk & Nishihara (Phys. Plasmas, vol. 4, 1997, 1028–1038) effectively describe the start-up and linear stages, respectively, across all cases. None of the considered nonlinear models is valid under all $A_1$ conditions. Based on the dependence of spike and bubble evolutions on $A_1$ provided by the present work and previous study (Chen et al., J. Fluid Mech., vol. 975, 2023, A29), the SEA model (Sadot et al., Phys. Rev. Lett., vol. 80, 1998, pp. 1654–1657), whose expression has clear physical meanings, is modified by revising the coefficient that governs its prediction for early-time evolution. The modified model applies to prediction of the weakly nonlinear evolution of RMI with $A_1$ ranging from −0.95 to −0.35 and from 0.30 to 0.86. Based on this model, an approximation of the critical $A_1$ for the occurrence of spike acceleration is obtained.

The thermocapillary flows generated by an inclined temperature gradient in and around a floating droplet are studied in the framework of the lubrication approximation. Numerical simulations of nonlinear flow regimes are fulfilled. It is shown that under the action of Marangoni stresses, a droplet typically moves as a whole. It is found that an inclined temperature gradient can lead to the excitation of periodic oscillations. With an increase of the inclination of the temperature gradient, temporally quasi-periodic oscillations have been obtained. In a definite region of parameters, an inclined temperature gradient can suppress oscillations, changing the droplet’s shape. The diagram of regimes in the plane of longitudinal and transverse Marangoni numbers has been constructed. Bistability has been found.

We report the first measurement of turbulent mixing developing from the convergent Richtmyer–Meshkov (RM) instability using time-resolved planar laser-induced fluorescence in a semi-annular convergent shock tube. A membraneless yet sharp interface with random short-wavelength perturbations, but controllable long-wavelength perturbations, is created by an automatically retractable plate, enhancing the reproducibility and reliability of RM turbulence experiments. The cylindrical air/SF$_6$ interface formed is first subjected to a convergent shock, then to its reflected shock and subsequently transits to turbulent mixing. It is found that the mixing width after reshock has a linear growth rate more than twice the rate in planar geometry. Also, the mixing width does not present power-law growth at late stages as in a planar geometry. However, the scalar spectrum and transition criterion obtained are similar to their planar counterparts. These findings indicate that the geometric constraint greatly affects the large scales of the flow, while having a weaker effect on the small scales. It is also found that the reflected shock significantly increases both scale separation and Reynolds number, explaining the rapid transition to turbulence following reshock.

Linear and nonlinear mechanisms governing the growth of second-mode waves are analysed using a newly derived disturbance energy conservation equation that highlights the physical processes responsible for fluctuation energy production, flux-based transport and destruction. Axisymmetric direct numerical simulations (DNS) data from a Mach 6 hypersonic boundary layer, simulated over a $3^\circ$ half-angle sharp cone at zero angle of attack, is used as a reference. A Legendre polynomial-based forcing methodology is used to trigger transition in the DNS over a range of various amplitude levels and different forcing frequency content. Closure of the disturbance energy budgets is demonstrated numerically using the DNS data. The terms responsible for the amplification of the disturbance are identified, and nonlinear attenuation effects are discussed. We show that the interaction between the entropy/velocity fluctuations and the base temperature gradient governs the second-mode growth in the linear regime. Energy production occurs in the critical layer due to non-isentropic processes and accumulates in acoustic form below the relative sonic line through downward transport. At higher forcing amplitudes, nonlinear spectral broadening is observed, with simultaneous thermoviscous diffusion attenuating the disturbance energy growth. This effect is responsible for the non-monotonic streamwise variation of the wall-pressure spectrum. Phase speed and growth rate analyses, informed by linear stability theory (LST), reveal wave steepening effects preceding this nonlinear attenuation effect. The disturbance energy is observed to match the LST predictions at lower forcing amplitudes, deviating, as expected, at higher amplitudes.

We find the optimally time-dependent (OTD) orthogonal modes about a time-varying flow generated by a strong gust vortex impacting a NACA 0012 airfoil. This OTD analysis reveals the amplification characteristics of perturbations about the unsteady base flow and their amplified spatiotemporal structures that evolve over time. We consider four time-varying laminar base flows in which a vortex with a strength corresponding to the gust ratio $G$ of $\{-1,-0.5,0.5,1\}$ impinges on the leading edge of the airfoil at an angle of attack of $12^\circ$. In these cases, the impingement of the strong gust vortex causes massive separation and the generation of large-scale vortices around the airfoil within two convective time units. As these flow structures develop around the airfoil on a short time scale, the airfoil experiences large transient vortical lift variations in the positive and negative directions that are approximately five to ten times larger than the baseline lift. The highly unsteady nature of these vortex–airfoil interactions necessitates an advanced analytical technique capable of capturing the transient perturbation dynamics. For each of the considered gust ratios, the OTD analysis identifies the most amplified region to perturbations, the location of which changes as the wake evolves differently. For interactions between a moderate positive vortex gust ($G=0.5$) and the airfoil, the area where perturbations are amplified transitions from the leading-edge vortex (LEV) sheet to the forming LEV. Later, this most amplified structure becomes supported in the airfoil wake directly behind the trailing edge. In contrast, a strong vortex gust ($G=\pm 1$) encountered by the airfoil shows the most amplified OTD mode to appear around the core of the shed vortices. This study provides an analysis technique and fundamental insights into the broader family of unsteady aerodynamic problems.

The nonlinear stability of two-dimensional (2-D) plane Couette flow subject to a constant throughflow is analysed at finite and asymptotically large Reynolds numbers $\textit {Re}$. The speed of this throughflow is quantified by the non-dimensional throughflow number $\eta$. The base flow exhibits a linear instability provided $\eta \gtrsim 3.35$, with multi-deck upper and lower branch structures developing in the limit $1\ll \eta \ll \mathit {O}(\textit {Re})$. This instability provides a springboard for the computation of nonlinear travelling waves which bifurcate subcritically from the linear neutral curve, allowing us to map out a neutral surface at different values of $\eta$. Using strongly nonlinear critical layer theory, we investigate the waves that bifurcate from the upper branch at asymptotically large $\textit {Re}$. This asymptotic structure exists provided the throughflow number is larger than the critical value of $\eta _c\approx 1.20$ and is shown to give quantitatively similar results to the numerical solutions at Reynolds numbers of $\mathit {O}(10^5)$.

When they occur, azimuthal thermoacoustic oscillations can detrimentally affect the safe operation of gas turbines and aeroengines. We develop a real-time digital twin of azimuthal thermoacoustics of a hydrogen-based annular combustor. The digital twin seamlessly combines two sources of information about the system: (i) a physics-based low-order model; and (ii) raw and sparse experimental data from microphones, which contain both aleatoric noise and turbulent fluctuations. First, we derive a low-order thermoacoustic model for azimuthal instabilities, which is deterministic. Second, we propose a real-time data assimilation framework to infer the acoustic pressure, the physical parameters, and the model bias and measurement shift simultaneously. This is the bias-regularized ensemble Kalman filter, for which we find an analytical solution that solves the optimization problem. Third, we propose a reservoir computer, which infers both the model bias and measurement shift to close the assimilation equations. Fourth, we propose a real-time digital twin of the azimuthal thermoacoustic dynamics of a laboratory hydrogen-based annular combustor for a variety of equivalence ratios. We find that the real-time digital twin (i) autonomously predicts azimuthal dynamics, in contrast to bias-unregularized methods; (ii) uncovers the physical acoustic pressure from the raw data, i.e. it acts as a physics-based filter; (iii) is a time-varying parameter system, which generalizes existing models that have constant parameters, and capture only slow-varying variables. The digital twin generalizes to all equivalence ratios, which bridges the gap of existing models. This work opens new opportunities for real-time digital twinning of multi-physics problems.

This paper presents numerical simulations of the free fall of homogenous cylinders of length-to-diameter ratios $2$, $3$ and 5 and solid-to-fluid-density ratios $\rho _s/\rho$ going from 0 to 10 in transitional regimes. The path instabilities are shown to be due to two types of transitional states. The well-known fluttering state is a solid mode, characterised by significant oscillations of the cylinder axis due to a strong interaction between the vortex shedding in the wake and the solid degrees of freedom. Weakly oscillating, mostly irregular trajectories, are fluid modes, associated with purely fluid instabilities in the wake. The interplay of solid and fluid modes leads to a varying scenario in which the length-to-diameter and density ratios play an important role. The description is accompanied by the presentation of the identified transitional states in terms of path characteristics and vorticity structure of the wakes and by bifurcation diagrams showing the evolution of asymptotic states with increasing Galileo numbers. There appears to be a strong difference between the behaviour of cylinders of aspect ratio $L/d=3$ and 5. A similar contrast is stated between light cylinders of density ratios $\rho _s/\rho \le 2$ and dense cylinders of density ratios 5 and 10. Finally, the question of the scatter of values of the drag coefficient and of the frequency of oscillations raised in the literature is addressed. It is shown, that in addition to external parameters (Galileo number, density and aspect ratio) the amplitude of oscillations characterising the instability development is to be taken into account to explain this scatter. Fits of the simulation results to simple correlations are proposed. Namely that of the drag coefficient proves to be accurate (better than 1 % of accuracy) but also that of the Strouhal number (a few per cent of accuracy) may be of practical use.

Rotor-stator flows are known to exhibit instabilities in the form of circular and spiral rolls. While the spirals are known to emanate from a supercritical Hopf bifurcation, the origin of the circular rolls is still unclear. In the present work we suggest a quantitative scenario for the circular rolls as a response of the system to external forcing. We consider two types of axisymmetric forcing: bulk forcing (based on the resolvent analysis) and boundary forcing using direct numerical simulation. Using the singular value decomposition of the resolvent operator the optimal response is shown to take the form of circular rolls. The linear gain curve shows strong amplification at non-zero frequencies following a pseudo-resonance mechanism. The optimal energy gain is found to scale exponentially with the Reynolds number $Re$ (for $Re$ based on the rotation rate and interdisc spacing $H$). The results for both types of forcing are compared with former experimental works and previous numerical studies. Our findings suggest that the circular rolls observed experimentally are the effect of the high forcing gain together with the roll-like form of the leading response of the linearised operator. For high enough Reynolds number it is possible to delineate between linear and nonlinear responses. For sufficiently strong forcing amplitudes, the nonlinear response is consistent with the self-sustained states found recently for the unforced problem. The onset of such non-trivial dynamics is shown to correspond in state space to a deterministic leaky attractor, as in other subcritical wall-bounded shear flows.

We conduct direct numerical simulations (DNS) to study the temporal and spatial developments of the roll waves on a laminar sheet flow of Newtonian fluid. The DNS unveil the physics of the wavefront and show the limitation of the widely used shallow-layer approximations. The most prominent wave, the front runner, is determined by the DNS for the first time in studying the spatial development of the laminar sheet flow with negligible surface tension. Depending on the Froude and Reynolds numbers, the front runner can be a multi-peaked undular bore or a single-peaked non-breaking or breaking wave. The simulation has uncovered an extended region behind the wavefront, where the bed-friction stress is much higher than the corresponding friction in the undisturbed uniform flow. It also produces an uplift velocity needed in the description of wave breaking. For comparison, we also examine the nonlinear development of the instability using two-equation and four-equation shallow-layer models. The two-equation shallow-layer model has produced the bulk of the wave profile but is deficient because it fails to predict the uplift velocity and the substantial increase in bed friction in the frontal region. The four-equation shallow-layer model correctly predicts the bed friction but cannot produce the breaking wave. The simulations also determine the celerity and amplitude of the front runner to follow a linear relationship, qualitatively similar to the roll waves in a turbulent flow.

The nonlinear waves in a sheared liquid film on a horizontal plate at small Reynolds numbers are examined by theoretical and numerical approaches. The analysis employs the long-wave approximation along with finite difference schemes. The results show that the surface tension can suppress disturbances and prevent the occurrence of singularities. While the film flow is driven by the shear stress on the interface, its instability highly depends on the magnitude and direction of gravity. Specifically, when the direction of gravity is opposite to the wall-normal direction, perturbations are stabilized by gravity. In contrast, when these two directions are the same, the gravitational force is destabilizing, and stationary travelling waves can exist if a balance is reached between the effects of gravity and surface tension. For the steady solitary waves, there are quasi-periodic oscillations occurring between two stationary points, indicating the presence of heteroclinic trajectories. For periodic waves, the evolutions are sensitive to several parameters and initial disturbances, while one steady-state wave exhibits a sine function-like behaviour.

In transonic flow conditions, buffeting associated with finite-amplitude lift fluctuations can limit the operational envelope of an aircraft. For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation. We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behaviour of the oscillatory buffet-onset instability on airfoils. The flow is governed by the unsteady Reynolds-averaged Navier–Stokes equations, with a basic state provided by a steady-state solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation. The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states. The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil. Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behaviour needed to estimate the buffet-onset boundary. The stability analysis is shown to capture the envelope of lift oscillations during a continuous pitch of an airfoil, from pre-buffet through post-buffet lift levels.

Understanding the generation of large-scale magnetic fields and flows in magnetohydro-dynamical (MHD) turbulence remains one of the most challenging problems in astrophysical fluid dynamics. Although much work has been done on the kinematic generation of large-scale magnetic fields by turbulence, relatively little attention has been paid to the much more difficult problem in which fields and flows interact on an equal footing. The aim is to find conditions for long-wavelength instabilities of stationary MHD states. Here, we first revisit the formal exposition of the long-wavelength linear instability theory, showing how long-wavelength perturbations are governed by four mean field tensors; we then show how these tensors may be calculated explicitly under the ‘short-sudden’ approximation for the turbulence. For MHD states with relatively little disorder, the linear theory works well: average quantities can be readily calculated, and stability to long-wavelength perturbations determined. However, for disordered basic states, linear perturbations can grow without bound and the purely linear theory, as formulated, cannot be applied. We then address the question of whether there is a linear response for sufficiently weak mean fields and flows in a dynamical (nonlinear) evolution, where perturbations are guaranteed to be bounded. As a preliminary study, we first address the nature of the response in a series of one-dimensional maps. For the full MHD problem, we show that in certain circumstances, there is a clear linear response; however, in others, mean quantities – and hence the nature of the response – can be difficult to calculate.

Self-sustained thermoacoustic oscillations as observed in low-emission combustion- involved gas turbines and aero-engines involve complicated thermal fluid–acoustics interaction and rich nonlinear dynamics. Such pulsating oscillations are known as thermoacoustic instability. When it occurs, large-amplitude limit cycle oscillations (LCOs) of thermodynamic parameters are frequently observed. These LCOs could cause overheating, flame flashback, and even engine failures. Thus it is critical to understand and predict the generation mechanisms and nonlinear dynamics behaviours, and then develop corresponding control approaches to prevent or control the onset of such instabilities. In this work, we develop and extend the classical van der Pol oscillators by integrating a physics-informed neural networks (PINNs) algorithm with a modelled nonlinear Rijke-type thermoacoustic combustor. The theoretical Rijke tube system (with Galerkin expansion and modified King's law implemented) and a CFD simulation model are applied to provide ‘training/calibration data’ for the extended van der Pol (EVDP)-PINNs model. The optimized EVDP oscillators are confirmed to be capable of capturing the key nonlinear characteristics by comparing the transient growth behaviours of thermodynamic perturbations and LCO amplitude and frequency. Further investigations are conducted to obtain Hopf bifurcation and amplitude death (AD) characteristics. Comparison is then made to the coupled EVDP systems. Quite similar Hopf bifurcation features, but differences in regions of AD, are observed. In general, we demonstrate an applicable approach to intelligently ‘learn’ a nonlinear thermoacoustic system and to create reliable EVDP oscillator systems, which have great potential to contribute to the development and testing of control approaches, such as the coupling described in this work, which may replace costly experimental tests.

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$-fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

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.