By analysing a database (Lozano-Durán & Jiménez, Phys. Fluids, vol. 26, 2014, 011702) of fully developed turbulent channel flow at the friction Reynolds number $Re_\tau =4179$, we investigate the sustaining mechanism of a hierarchy of coherent structures in the turbulence. For this purpose, we decompose the turbulent fields into different scales by band-pass filters and quantify the real-space energy transfer. Visualizations of the hierarchy of vortices and velocity in the filtered fields show that the largest-scale structures at each distance from the wall are composed of quasi-streamwise vortices and low-speed streaks. These are similar to well known coherent structures in the buffer layer and they are maintained by a hierarchical self-sustaining process. Quantitatively, however, the energy production rate of the largest-scale structures is different in the log and buffer layers. This difference explains the change of the scaling of the Reynolds stress as a function of the Reynolds number. In contrast to the largest-scale structures, vortices smaller than the distance from the wall distribute isotropically, and they are generated by an energy cascading process. The energy of these small-scale structures is transferred predominantly from twice-larger-scale structures and reduced by half-scale ones through the vortex stretching and contraction, respectively. Turbulent advection from the wall hardly contributes to the maintenance of small-scale structures in the log layer.

This study investigates the transitional wake dynamics of a simultaneously pitching– heaving airfoil in the low Reynolds number regime. The transition from Kármán to reverse Kármán wakes and the subsequent wake deflection are known to take place entirely in the periodic regime as the dynamic heave velocity $\kappa h$ (proportionally, the amplitude-based Strouhal number $St_A$) is gradually increased. However, further increase in $\kappa h$ may result in the loss of dynamic stability of the deflected vortex street and an eventual transition to chaos. This phenomenon has largely remained unexplored in the existing literature. The present study attempts to fill this gap and aims to establish a dynamic link between the near- and far-field wake transitions. It is shown that the deflected jet undergoes a switching of deflection direction in the far-field as the near-field encounters a quasi-periodic transition with the increase in $\kappa h$. The quasi-periodic behaviour of the near-field is seen to get interspersed with intermittent aperiodic windows, resulting in a complete reversal of the deflection direction through flipping of the immediate vortex couple. Eventually, the wake topology becomes fully chaotic through a series of rapid aperiodic jet-switching. To the best of the authors’ knowledge, this is the first study that investigates the role of this aperiodic jet-switching in ushering in chaos in the wake of a pitch–heave flapping system. The mechanism of jet-switching and the role of leading-edge vortex are also revealed. Flow field dynamics during these changes and the underlying vortex interactions are analysed using nonlinear dynamical tools.

The dynamics of the merging of two distinct holes (hole–hole) and a single hole with a straight rim (hole–rim) are investigated using three-dimensional numerical simulations. Thin liquid sheets with thicknesses ranging from $25$ to $150\ \mathrm {\mu }\textrm {m}$ are considered using air/water conditions. Preliminary simulations of a single hole expansion and the retraction of a bounded liquid sheet edge have shown good agreements with the well-known Taylor–Culick end rim regime. For the hole–hole and the hole–rim cases, our computations reveal that the liquid bridge, formed after the merging, is subjected to three different motions: the extension of the bridge, the mid-plane contraction and vertical/horizontal oscillations of its diameter, with an exponentially decaying amplitude. The mid-plane diameter of the liquid bridge is shown to decrease linearly with time for the hole–rim case while a faster quadratic decrease is observed for the hole–hole case. The small ratio of the extension rate to the capillary contraction rate indicates a slow extension limit of the bridge. However, the liquid bridge does not contract and pinch off on a single point, as predicted in the literature, and its central part forms a cylindrical ligament. Thus, this cylindrical part pinches off at its ends and forms a receding ligament with bulbous ends which can either detach by the end-pinching mechanism or remain attached, recoil and merge into a single large drop. The size of the formed drop, in the case where the ligament does not break, is later expressed as a power-law function of the initial liquid sheet thickness.

In the presence of inertia-gravity waves, the geostrophic and hydrostatic balance that characterises the slow dynamics of rapidly rotating, strongly stratified flows holds in a time-averaged sense and applies to the Lagrangian-mean velocity and buoyancy. We give an elementary derivation of this wave-averaged balance and illustrate its accuracy in numerical solutions of the three-dimensional Boussinesq equations, using a simple configuration in which vertically planar near-inertial waves interact with a barotropic anticylonic vortex. We further use the conservation of the wave-averaged potential vorticity to predict the change in the barotropic vortex induced by the waves.

Incompressible fluid flows are characterised by high correlations between low pressure and high velocity and vorticity. The velocity–pressure correlation is an immediate consequence of fluid acceleration towards low-pressure regions. On the other hand, fluid converging to a low-pressure centre is driven sideways by a resistance due to incompressibility, giving rise to the formation of a strong vortex, hence the vorticity–pressure correlation. Meanwhile, the formation of such a vortex effectively shields the low-pressure centre from incoming energetic fluid. As a result, a local pressure minimum can usually be found at the centre of a vortex where the vorticity is greatest but the velocity is relatively low, hence the misalignment of local pressure minima and velocity maxima. For Navier–Stokes flows, this misalignment has profound implications for extreme momentum growth and maintenance of regularity. This study examines the role of the velocity–pressure correlation in the problem of Navier–Stokes global regularity. On the basis of estimates for flows locally satisfying the critical scaling of the Navier–Stokes system, a qualitative theory of this correlation is considered. The theory appears to be readily quantified, advanced and tested by theoretical, mathematical and numerical methods. Regularity criteria depending on the degree of the velocity–pressure correlation are presented and discussed in light of this theory. The result suggests that as long as global pressure minimum (or minima) and velocity maximum (or maxima) are mutually exclusive, then regularity is likely to persist. This is the first result that makes use of an explicit measure of the velocity–pressure correlation as a key factor in the maintenance of regularity or development of singularity.

A theoretical and numerical study of complex sliding flows of yield-stress fluids is presented. Yield-stress fluids are known to slide over solid surfaces if the tangential stress exceeds the sliding yield stress . The sliding may occur due to various microscopic phenomena such as the formation of an infinitesimal lubrication layer of the solvent and/or elastic deformation of the suspended soft particles in the vicinity of the solid surfaces. This leads to a ‘stick–slip’ law which complicates the modelling and analysis of the hydrodynamic characteristics of the yield-stress fluid flow. In the present study, we formulate the problem of sliding flow beyond one-dimensional rheometric flows. Then, a numerical scheme based on the augmented Lagrangian method is presented to attack these kind of problems. Theoretical tools are developed for analysing the flow/no-flow limit. The whole framework is benchmarked in planar Poiseuille flow and validated against analytical solutions. Then two more complex physical problems are investigated: slippery particle sedimentation and pressure-driven sliding flow in porous media. The yield limit is addressed in detail for both flow cases. In the particle sedimentation problem, method of characteristics – slipline method – in the presence of slip is revisited from the perfectly plastic mechanics and used as a helpful tool in addressing the yield limit. Finally, flows through model and randomized porous media are studied. The randomized configuration is chosen to capture more sophisticated aspects of the yield-stress fluid flows in porous media at the yield limit – channelization.

We investigate aspects of the spherical squirmer model employing both large-scale numerical simulations and asymptotic methods when the squirmer is placed in weakly elastic fluids. The fluids are modelled by differential equations, including the upper-convected Maxwell (UCM)/Oldroyd-B, finite-extensibility nonlinear elastic model with Peterlin approximation (FENE-P) and Giesekus models. The squirmer model we examine is characterized by two dimensionless parameters related to the fluid velocity at the surface of the micro-swimmer: the slip parameter $\xi $ and the swirl parameter $\zeta $. We show that, for swimming in UCM/Oldroyd-B fluids, the elastic stress becomes singular at a critical Weissenberg number, Wi, that depends only on $\xi$. This singularity for the UCM/Oldroyd-B models is independent of the domain exterior to the swimmer, or any other forces considered in the momentum balance for the fluid – we believe that this is the first time such a singularity has been explicitly demonstrated. Moreover, we show that the behaviour of the solution at the poles is purely extensional in character and is the primary reason for the singularity in the Oldroyd-B model. When the Giesekus or the FENE-P models are utilized, the singularity is removed. We also investigate the mechanism behind the speed and rotation rate enhancement associated with the addition of swirl in the swimmer's gait. We demonstrate that, for all models, the speed is enhanced by swirl, but the mechanism of enhancement depends intrinsically on the rheological model employed. Special attention is paid to the propulsive role of the pressure and elucidated upon. We also study the region of convergence of the perturbation solutions in terms of Wi. When techniques that accelerate the convergence of series are applied, transformed solutions are derived that are in very good agreement with the results obtained by simulations. Finally, both the analytical and numerical results clearly indicate that the low-Wi region is more important than one would expect and demonstrate all the major phenomena observed when swimming with swirl in a viscoelastic fluid.

We present a numerical study analysing the breakup of a single cohesive particle aggregate in turbulence. Solid particles with diameters smaller than the Kolmogorov length scale ($d_p<\eta$) are initially aggregated into a spherical ‘clump’ of diameter $D>\eta$ and placed in homogeneous isotropic turbulence. Parameters are chosen relevant to dust or powder suspended in air such that cohesion due to van der Waals is important. Simulations are performed using an Eulerian–Lagrangian framework that models two-way coupling between the fluid and solid phases and resolves particle–particle interactions. Aggregate breakup is investigated for different adhesion numbers ${Ad}$, Taylor microscale Reynolds numbers ${Re}_\lambda$ and non-dimensional clump sizes $D/d_p$. The intermittency of turbulence is found to play a key role on the early stage breakup process, which can be characterized by a turbulent adhesion number ${Ad}_\eta$ that relates the potential energy of the van der Waals force to turbulent shear stresses. A scaling analysis shows that the time rate of breakup for each case collapses when scaled by ${Ad}_\eta$ and an aggregate Reynolds number proportional to $D$. A phenomenological model of the breakup process is proposed that acts as a granular counterpart to the Taylor analogy breakup (TAB) model commonly used for droplet breakup. Such a model is useful for predicting particle breakup in coarse-grained simulation frameworks, such as Reynolds-averaged Navier–Stokes, where relevant spatial and temporal scales are not resolved.

We present the first detailed thermal and velocity field characterization of convection in a rotating cylindrical tank of liquid gallium, which has thermophysical properties similar to those of planetary core fluids. Our laboratory experiments, and a closely associated direct numerical simulation, are all carried out in the regime prior to the onset of steady convective modes. This allows us to study the oscillatory convective modes, sidewall modes and broadband turbulent flow that develop in liquid metals before the advent of steady columnar modes. Our thermo-velocimetric measurements show that strongly inertial, thermal wind flows develop, with velocities reaching those of non-rotating cases. Oscillatory bulk convection and wall modes coexist across a wide range of our experiments, along with strong zonal flows that peak in the Stewartson layer, but that extend deep into the fluid bulk in the higher supercriticality cases. The flows contain significant time-mean helicity that is anti-symmetric across the midplane, demonstrating that oscillatory liquid metal convection contains the kinematic components to sustain system-scale dynamo generation.

The present study focuses on acoustically induced thermal effects on Rayleigh streaming inside a resonator. Firstly, we consider the effect of the transverse (or wall-normal) mean temperature gradient on the acoustic streaming flow generated by a standing wave between two parallel plates. Analytical expressions for acoustic quantities are developed and used to express the sources of linear streaming. The influence of a transverse temperature variation on the streaming velocity is clearly identified through a term proportional to the temperature difference and to the square of the half-width of the guide. This term modifies the Rayleigh streaming pattern and may generate an additional vortex. On the other hand, the longitudinal (or wall-parallel) temperature difference is calculated as a cumulated effect of thermoacoustic heat transport in the fluid, heat conduction in the wall and heat convection of the air outside the resonator. At high acoustic levels, heat is significantly convected by the streaming flow and the resulting transverse temperature difference is proportional to the longitudinal temperature difference. Combining these expressions brings out a new criterion parameter for the nonlinear Reynolds number ($Re_{NL}$) characterizing the transition in streaming patterns at high acoustic levels. This result explains previous experimental and numerical observations of the streaming flow dynamics at high acoustic amplitudes, under different temperature boundary conditions, and can provide a powerful prediction tool for streaming pattern transitions.

The spatial development of a turbulent boundary layer (TBL) subjected to freestream turbulence (FST) is investigated experimentally in a water channel for friction Reynolds numbers up to $Re_{\tau }=5060$. Four different FST intensities are generated with an active grid, ranging from a low-turbulence reference case to $u^{\prime }_{\infty }/U_{\infty }=12.5\,\%$. Wall-normal velocity scans are performed with laser doppler velocimetry at three positions downstream of the grid. There are two combating influences as the flow develops: the TBL grows while the FST decays. Whilst previous studies have shown the wake region of the TBL is suppressed by FST, the present measurements demonstrate that the wake recovers sufficiently far downstream. For low levels of FST, the near-wall variance peak grows as one moves downstream, whereas high FST results in an initially high variance peak that decays with streamwise position. These results are mirrored in the evolution of the spectrograms, where low FST results in the emergence of an outer spectral peak as the flow evolves, while high FST sees an initially high outer spectral peak decay in space. This finding is significant as it suggests the FST does not permanently mature the TBL ahead of its natural evolution. Finally, it is explicitly demonstrated that it is not sufficient to characterize the TBL solely by conventional parameters such as $Re_{\tau }$, but that the level of FST and the evolution of the two flows must also be considered.

Cylindrical vesicle and cell membranes under tension can undergo a Rayleigh–Plateau instability leading to break-up. In Part 1 (Graessel et al., J. Fluid Mech., vol. xxx, 2021, Ax) we showed that anisotropic tension, created by active biological processes underneath the cell membrane, can significantly influence this process for a liquid–liquid interface. Here, we study the combined influence of anisotropic tension and membrane elasticity on the Rayleigh–Plateau instability. We analytically derive the dispersion relation for an interface endowed with bending and/or shear elasticity considering explicitly the dynamics of the suspending fluid. We find that the combination of bending elasticity and tension anisotropy leads to three qualitatively different regimes for the Rayleigh–Plateau scenario: (i) the classical regime in which short wavelengths are stable and long wavelengths are unstable, (ii) the suppressed regime in which the system is stable against all perturbation wavelengths and (iii) the restricted regime, in which a stable region at short and another one at long wavelengths are separated by a range of unstable modes centred around the dimensionless wavenumber $kR_0=1$. The width of this unstable range as well as the dominant wavelength of the instability depend on the bending modulus and tension anisotropy. For shear elasticity and area dilatation, on the other hand, only the classical and the suppressed regimes are observed, with the transition between them being independent of the tension anisotropy.

We present an Euler–Lagrange approach for simulating magneto-Archimedes separation of almost neutrally buoyant spherical particles in the flow of a paramagnetic liquid, which is of direct relevance for separating different types of plastic by magnetic density separation. A four-way coupled point-particle method is employed where all relevant interactions between an external magnetic field, a magnetic fluid and discrete immersed particles are taken into account. Particle–particle interaction is modelled by a hard-sphere collision model which takes the interstitial fluid effects into account. First, the motion of rigid spherical particles in a paramagnetic liquid is studied in single- and two-particle systems. We find good agreements between our numerical results and experiments performed in a paramagnetic liquid exposed to a non-homogeneous magnetic field, also in the case of two colliding particles. Next, we investigate the magneto-Archimedes separation of particles with different mass densities in many-particle systems interacting with the fluid. Our results reveal that history effects and interparticle interactions significantly influence the levitation dynamics of particles and have a detrimental impact on the separation performance. We also investigate the effect of particle size and initial distribution on the separation performance. Results show that a reduction in the particle size from 4 to 2 mm leads to a $40\,\%$ increase in the separation time. Moreover, preseparation of particles into two groups of light and heavy particles decreases the separation time by $33\,\%$. The presented method is shown to be a robust and efficient computational framework for the investigation of particle-laden flows of magnetically responsive fluids.

Numerous experiments and theoretical calculations have shown that cylindrical vesicles can undergo a pearling instability similar to the Rayleigh–Plateau instability of a liquid jet when they are subjected to external tension. In a living cell, a Rayleigh–Plateau-like instability could be triggered by internal tension generated in the cell cortex. This mechanism has been suggested to play an essential role in biological processes such as cell morphogenesis. In contrast to the simple, passive and isotropic membrane of vesicles, the cortical tensions generated by biological cells are often strongly anisotropic. Here, we theoretically investigate how this anisotropy affects the Rayleigh–Plateau instability mechanism. We do so in the limit of both low and high Reynolds numbers and accordingly cover cell behaviour under anisotropic cortical tension as well as fast liquid jets with anisotropic surface tension. Combining analytical linear stability analysis with numerical simulations we report a strong influence of the anisotropy on the dominant wavelength of the instability: increasing azimuthal with respect to axial tension leads to destabilisation and to a shorter break-up wavelength. In addition, compared to the classical isotropic Rayleigh–Plateau situation, the range of unstable modes grows or shrinks when the azimuthal tension is higher or lower than the axial tension, respectively. We explore nonlinear effects like an altered break-up time and formation of satellite droplets under anisotropic tension. In Part 2 (Bächer et al. J. Fluid Mech., vol. xxx, 2021, Ax) of this series we continue our analysis by analytically investigating the influence of bending and shear elasticity, usually present in vesicles and cells, on this anisotropic Rayleigh–Plateau instability.

One of the key observations in the Princeton Superpipe was the late start of the logarithmic mean velocity overlap layer at a wall distance of the order of $10^3$ inner units. Between $y^+\approx 150$, the start of the overlap layer in zero pressure gradient turbulent boundary layers, and $y^+\approx 500$, the Superpipe profile is modelled equally well by a power law or a log law with a larger slope than in the overlap layer. This paper demonstrates, that the asymptotic mean velocity profile in turbulent plane channel flow exhibits analogous characteristics, namely a rather sudden decrease of logarithmic slope (increase of $\kappa$) at a $y^+$ of approximately $600$, which marks the start of the actual overlap layer. The demonstration results from the first construction of the complete mean velocity inner and outer asymptotic expansions up to order ${O}({Re}^{-1}_{\tau })$ from direct numerical simulations (DNS) at moderate Reynolds numbers. The ${O}({Re}^{-1}_{\tau })$ contribution to the indicator function $\varXi ^+ = y^+(\textrm {d} U^+/ \textrm {d} y^+)$ is found to be important and to prevent the direct determination of $\kappa$ from currently available channel DNS. A preliminary, leading-order analysis of a Couette flow DNS, on the other hand, yields an increase of logarithmic slope (decrease of $\kappa$) at a $y^+_{{break}}\approx 400$. The correlation between the sign of the slope change and the flow symmetry motivates the hypothesis that the breakpoint between the possibly universal short inner logarithmic region and the actual overlap log-law corresponds to the penetration depth of large-scale turbulent structures originating from the opposite wall.

The post-injection migration of a plume of CO$_{2}$ through an inclined, confined porous layer across which the permeability varies is studied theoretically. We derive a sharp-interface lubrication model which accounts for the capillary trapping of CO$_{2}$ at the receding edge of the plume. Eventually the CO$_{2}$ becomes entirely trapped in the pore throats, and the final run-out distance is a key quantity for determining storage security and efficiency. We deploy asymptotic approximations to show that when the CO$_{2}$ plume migrates a long distance relative to its initial length, the run-out distance is approximately proportional to the permeability at the top of the layer. The permeability structure away from the top boundary is important at early and intermediate times. Dissolution of the CO$_{2}$ and three-dimensional effects are incorporated, which demonstrate that the influence of heterogeneity is quite general. The initial distribution of the CO$_{2}$ at the end of the injection phase also influences the post-injection spreading and trapping. At low injection rates, the CO$_{2}$ remains near the top of the layer so that the end-of-injection plume shape has a small aspect ratio leading to a relatively large run-out length. At very high fluxes, the end-of-injection shape and hence the final run-out distance becomes nearly independent of the injection flux because the role of buoyancy becomes negligible during injection. Our results illustrate the strong control of reservoir heterogeneity on the sweep efficiency of a CO$_{2}$ plume and hence the storage capacity. In many situations, it may be possible to increase the storage volume by injecting at a higher rate.

Continuously forced, stratified shear flows occur in many geophysical systems, including flow over sills, through fjords and at the mouths of rivers and estuaries. These continuously forced shear flows can be unstable and drive turbulence, which can enhance the rate of mixing. In this study, we analyse three-dimensional direct numerical simulations of an idealized stratified shear flow that is continuously forced by weakly relaxing both the buoyancy and streamwise velocity towards prescribed mean profiles. We explore a range of large and small Richardson numbers, for constant Reynolds and Prandtl numbers (${Re}=4000$ and ${Pr}=1$). After a turbulent steady state develops, three regimes are observed: (i) a weakly stratified, overturning regime, (ii) a strongly stratified, scouring regime and (iii) an intermediately stratified, intermittent regime. The overturning regime exhibits partially formed overturning billows that break down into turbulence and broaden the velocity and buoyancy interfaces. Conversely, the scouring regime exhibits internal gravity waves propagating along the strongly stratified buoyancy interface, while turbulence on either side of the buoyancy interface reinforces the stratification. The intermediate regime quasi-periodically alternates between behaviours associated with the overturning and scouring regimes. For each case, we quantify an appropriate measure of the efficiency of mixing and examine its dependence on relevant parameters including appropriate definitions of the buoyancy Reynolds number, gradient Richardson number and horizontal Froude numbers. Using a framework involving sorted buoyancy coordinates as introduced by Nakamura (J. Atmos. Sci., vol. 53, 1996, pp. 1524–1537) and Winters & D'Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193), we examine the underlying physical mechanisms leading to broadening and thinning of the buoyancy interface.

The propulsive performance of a flexible foil with prescribed pitching and heaving motions about any pivot point location and passive chordwise flexural deflection is analysed within the framework of the linear potential flow theory and the Euler–Bernoulli beam equation using a quartic approximation for the deflection. The amplitude of the flexural component of the deflection and its phase, the thrust force, input power and propulsive efficiency are computed analytically in terms of the stiffness and mass ratio of the plate, frequency, pivot point location and remaining kinematic parameters. It is found that the maximum flexural deflection amplitude, thrust and input power are related to the first fluid–structure natural frequency of the system, corresponding to the deflection approximation considered. The same relation is observed for the propulsive efficiency when an offset drag is included in the analytical expressions. These results, which are valid for small amplitude and sufficiently large stiffness of the foil, are compared favourably with previous related results when the foil pivots about the leading edge. The configurations generating maximum thrust and efficiency enhancement by flexibility are analysed in relation to those of an otherwise identical rigid foil.

In turbulent wall sheared thermal convection, there are three different flow regimes, depending on the relative relevance of thermal forcing and wall shear. In this paper, we report the results of direct numerical simulations of such sheared Rayleigh–Bénard convection, at fixed Rayleigh number $Ra=10^{6}$, varying the wall Reynolds number in the range $0 \leqslant Re_w \leqslant 4000$ and Prandtl number $0.22 \leqslant Pr \leqslant 4.6$, extending our prior work by Blass et al. (J. Fluid Mech., vol. 897, 2020, A22), where $Pr$ was kept constant at unity and the thermal forcing ($Ra$) varied. We cover a wide span of bulk Richardson numbers $0.014 \leqslant Ri \leqslant 100$ and show that the Prandtl number strongly influences the morphology and dynamics of the flow structures. In particular, at fixed $Ra$ and $Re_w$, a high Prandtl number causes stronger momentum transport from the walls and therefore yields a greater impact of the wall shear on the flow structures, resulting in an increased effect of $Re_w$ on the Nusselt number. Furthermore, we analyse the thermal and kinetic boundary layer thicknesses and relate their behaviour to the resulting flow regimes. For the largest shear rates and $Pr$ numbers, we observe the emergence of a Prandtl–von Kármán log layer, signalling the onset of turbulent dynamics in the boundary layer.

We study the effect of the Coriolis force on centrifugal buoyancy-driven convection in a rotating cylindrical shell with inner cold wall and outer hot wall. This is done by performing direct numerical simulations for increasing inverse Rossby number $Ro^{-1}$ from zero (no Coriolis force) to $20$ (very large Coriolis force) and for Rayleigh number $Ra$ from $10^{7}$ to $10^{10}$ and Prandtl number $Pr = 0.7$, corresponding to air. We invoke the thin-shell limit, which neglects the curvature and radial variations of the centripetal acceleration. As $Ro^{-1}$ increases from zero, the system forms an azimuthal bidirectional wind that reaches its maximum momentum at an optimal $Ro^{-1}_{opt}$, associated with a maximal skin-friction coefficient $C_f$ and a minimal Nusselt number $Nu$. Just beyond $Ro^{-1}_{opt}$, the wind weakens and an axial, quasi-two-dimensional cyclone, corotating with the system, begins to form. A local ‘turbulence’ inverse Rossby number (non-dimensionalised by the eddy turnover time) determines the onset of cyclone formation for all $Ra$, when its value reaches approximately $4$. At $Ro^{-1} \gg Ro^{-1}_{opt}$, the system falls into the geostrophic regime with a sudden drop in $Nu$. The bidirectional wind for $Ro^{-1} \le Ro^{-1}_{opt}$ is a feature of this system, as it hastens the boundary layer transition from laminar to turbulent, towards the ultimate regime. We see the onset of this transition at $Ra=10^{10}$ and $Ro^{-1}\simeq Ro^{-1}_{opt}$, although the mean flow profile has not yet fully collapsed on the Prandtl–von Kármán (logarithmic) law.