We study the axisymmetric spreading of drops deposited on a pre-existing horizontal layer of the same viscous fluid. Using a combination of experiments, numerical modelling based on the axisymmetric free-surface Navier–Stokes equations and scaling analyses, we explore the drops’ behaviour in a regime where the flow is driven by gravitational and/or capillary forces while inertial effects are small. We find that during the early stages of the drops’ evolution there are three distinct spreading behaviours depending on the thickness of the liquid layer. For thin layers the fluid ahead of a clearly defined spreading front is at rest and the overall behaviour resembles that of a drop spreading on a dry substrate. For thicker films, the spreading is characterised by an advancing wedge which is sustained by fluid flow from the drop into the layer. Finally, for thick layers the drop sinks into the layer, accompanied by significant flow within the layer. As the drop keeps spreading, the evolution of its shape becomes self-similar, with a power-law behaviour for its radius and its excess height above the undisturbed fluid layer. We employ lubrication theory to analyse the drop’s ultimate long-term behaviour and show that all drops ultimately enter an asymptotic regime which is reached when their excess height falls below the thickness of the undisturbed layer.

This paper presents a numerical study of the sound generated by turbulent, premixed flames. Direct numerical simulations (DNS) of two round jet flames with equivalence ratios of 0.7 and 1.0 are first carried out. Single-step chemistry is employed to reduce the computational cost, and care is taken to resolve both the near and far fields and to avoid noise reflections at the outflow boundaries. Several significant features of these two flames are noted. These include the monopolar nature of the sound from both flames, the stoichiometric flame being significantly louder than the lean flame, the observed frequency of peak acoustic spectral amplitude being consistent with prior experimental studies and the importance of so-called ‘flame annihilation’ events as acoustic sources. A simple model that relates these observed annihilation events to the far-field sound is then proposed, demonstrating a surprisingly high degree of correlation with the far-field sound from the DNS. This model is consistent with earlier works that view a premixed turbulent flame as a distribution of acoustic sources, and provides a physical explanation for the well-known monopolar content of the sound radiated by premixed turbulent flames.

Plane Poiseuille flow has long served as the simplest testing ground for Tollmien–Schlichting wave instability. In this paper, we provide a comprehensive comparison of equilibrium Tollmien–Schlichting wave solutions arising from new high-resolution Navier–Stokes calculations and the corresponding predictions of various large-Reynolds-number asymptotic theories developed in the last century, such as double-deck theory, viscous nonlinear critical layer theory and strongly nonlinear critical layer theory. In the relatively small to moderate amplitude regime, the theories excellently predict the behaviour of the numerical solutions at Reynolds numbers of order $10^{6}$ and above, whilst for larger amplitudes our computations suggest the need for further asymptotic theories to be developed.

In this paper, we have considered the effects of the shallowness of the domain as well as the air–water free surface on the stratified shear instabilities of the fluid underneath. First, we numerically solve the non-Boussinesq Taylor–Goldstein equation for smooth velocity and density profiles of a model shear layer with a free surface. When the depth of the fluid is relatively shallow compared to the shear layer thickness, the surface gravity waves existing at the free surface come closer to the waves existing in the shear layer. This can lead to resonant wave interactions, making the flow unstable to more varieties of modal instabilities. In order to obtain a deeper understanding of the instability mechanisms, we have performed analytical studies with broken-line profiles (profiles for which vorticity and density are piecewise constant). Furthermore, reduced-order broken-line profiles have also been developed, based on which dispersion diagrams are constructed. Through these diagrams we have underpinned the resonantly interacting waves leading to each type of instability. Two new instabilities have been found; one of them, referred to as the ‘surface gravity – interfacial gravity (SG-IG) mode’, arises due to the interaction between a surface gravity wave and an interfacial gravity wave, and would therefore be absent if there is no internal density stratification. The other one – the ‘surface gravity – lower vorticity (SG-LV) mode’, which arises due to the interaction between a surface gravity wave and the lower vorticity wave, surpasses Kelvin–Helmholtz (KH) instability to become the most unstable mode, provided the system is significantly shallow. Stability boundary of the SG-LV mode is found to be quite different from that of KH. In fact, KH becomes negligible for relatively shallow flows, while SG-LV’s growth rate is significant – comparable to the growth rate of KH for unbounded domains (${\approx}0.18$). Moreover, the SG-LV mode is found to be analogous to the barotropic mode observed in two-layer quiescent flows. We have found that the effect of a free surface on the Holmboe instability is not appreciable. Holmboe in the presence of a free surface is found to be analogous to the baroclinic mode observed in two-layer quiescent flows. Except for the Holmboe instability, remarkable differences are observed in all other instabilities occurring in shallow domains when the air–water interface is replaced by a rigid lid. We infer that the rigid-lid approximation is valid for large vertical domains and should be applied with caution otherwise. Furthermore, we have also shown that if shear is absent at the free surface, our problem can be modelled using Boussinesq approximation, that is, $O(1)$ density variations in the inertial terms can still be neglected.

We conduct minimal-channel direct numerical simulations of turbulent flow over two-dimensional rectangular bars aligned in the spanwise direction. This roughness has often been described as $d$-type, as the roughness function $\unicode[STIX]{x0394}U^{+}$ is thought to depend only on the outer-layer length scale (pipe diameter, channel half-height or boundary layer thickness). This is in contrast to conventional engineering rough surfaces, named $k$-type, for which $\unicode[STIX]{x0394}U^{+}$ depends on the roughness height, $k$. The minimal-span rough-wall channel is used to circumvent the high cost of simulating high Reynolds number flows, enabling a range of bars with varying aspect ratios to be investigated. The present results show that increasing the trough-to-crest height, $k$, of the roughness while keeping the width between roughness bars, ${\mathcal{W}}$, fixed in viscous units, results in non-$k$-type behaviour although this does not necessarily indicate $d$-type behaviour. Instead, for deep surfaces with $k/{\mathcal{W}}\gtrsim 3$, the roughness function appears to depend only on ${\mathcal{W}}$ in viscous units. In these situations, the flow no longer has any information about how deep the roughness is and instead can only ‘see’ the width of the fluid gap between the bars.

The scaling of the turbulent/non-turbulent interface (TNTI) at high Reynolds numbers is investigated by using direct numerical simulations (DNS) of temporal turbulent planar jets (PJET) and shear free turbulence (SFT), with Reynolds numbers in the range $142\leqslant Re_{\unicode[STIX]{x1D706}}\leqslant 400$. For $Re_{\unicode[STIX]{x1D706}}\gtrsim 200$ the thickness of the TNTI ($\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}$), like that of its two sublayers – the viscous superlayer (VSL, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}$) and the turbulent sublayer (TSL, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}$) – all scale with the Kolmogorov micro-scale $\unicode[STIX]{x1D702}$, while the particular scaling constant depends on the sublayer. Specifically, for $Re_{\unicode[STIX]{x1D706}}\gtrsim 200$ while the VSL is always of the order of $\unicode[STIX]{x1D702}$, with $4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D708}}\rangle /\unicode[STIX]{x1D702}\leqslant 5$, the TSL and the TNTI are typically equal to $10\unicode[STIX]{x1D702}$, with $10.4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}\rangle /\unicode[STIX]{x1D702}\leqslant 12.5$, and $15.4\leqslant \langle \unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D714}}\rangle /\unicode[STIX]{x1D702}\leqslant 16.8$, respectively.

The unsteady three-dimensional flow organization of jets issued from a duct with swirl vanes at Reynolds number equal to 1000 and swirl number $S$ ranging between 0 and 0.8 is investigated. Time-resolved tomographic particle image velocimetry returns the instantaneous flow structure and its evolution by visualization of velocity and vortical features. The most relevant coherent motions are identified and characterized with the aid of dynamic mode decomposition. The time-averaged flow topology indicates that the vanes used to impart the swirling motion have a significant impact on the azimuthal modulation of momentum, with the jet exhibiting four sectors separated by a thin cross-like wake resulting from the boundary layer developed along the vane walls. The flow field is thus characterized by inner and outer shear regions. An increase in swirl, up to moderate levels ($S=0.4$), causes larger jet spreading angles. Further increase of the swirl number is accompanied by the appearance of a central recirculation zone due to vortex breakdown at $S=0.6$ which increases in size and is triggered upstream for increasing $S$. Although no shear layer instability development is observed at $S=0$, already at $S=0.2$ the swirling motion promotes the growth of helical vortices appearing as Kelvin–Helmholtz waves that deform the outer axial shear layer. The downstream evolution features successive pairing, which is observed for all the considered swirl numbers. The initial development of the instability is independent for each vane, whereas a mutual interaction between the vanes occurs after the vortex pairing. The reconnection from the four sectors vortices induces a significant increase of azimuthal vorticity, which affects the dynamical behaviour of the precessing vortex core. The latter is visualized by a low-order spatio-temporal reconstruction based on few dynamical modes. At a higher swirl number ($S\geqslant 0.6$), the axial vorticity component dominates the flow field; it interacts with the azimuthal vorticity, which penetrates inward through the meanders of the vane wakes and forces the vortex core precession and breakdown.

The spherical Couette system, consisting of a viscous fluid between two differentially rotating concentric spheres, is studied using numerical simulations and compared with experiments performed at BTU Cottbus-Senftenberg, Germany. We concentrate on the case where the outer boundary rotates fast enough for the Coriolis force to play an important role in the force balance, and the inner boundary rotates slower or in the opposite direction as compared to the outer boundary. As the magnitude of differential rotation is increased, the system is found to transition through three distinct hydrodynamic regimes. The first regime consists of the emergence of the first non-axisymmetric instability. Thereafter one finds the onset of ‘fast’ equatorially antisymmetric inertial modes, with pairs of inertial modes forming triadic resonances with the first instability. A further increase in the magnitude of differential rotation leads to the flow transitioning to turbulence. Using an artificial excitation, we study how the background flow modifies the inertial mode frequency and structure, thereby causing departures from the eigenmodes of a full sphere and a spherical shell. We investigate triadic resonances of pairs of inertial modes with the fundamental instability. We explore possible onset mechanisms through numerical experiments.

Compressible flow varies from ideal-gas behaviour at high pressures where molecular interactions become important. It is widely accepted that density is well described through a cubic equation of state while enthalpy and sound speed are functions of both temperature and pressure, based on two parameters, $A$ and $B$, related to intermolecular attraction and repulsion, respectively. Assuming small variations from ideal-gas behaviour, a closed-form approximate solution is obtained that is valid over a wide range of conditions. An expansion in these molecular interaction parameters simplifies relations for flow variables, elucidating the role of molecular repulsion and attraction in variations from ideal-gas behaviour. Real-gas modifications in density, enthalpy and sound speed for a given pressure and temperature lead to variations in many basic compressible-flow configurations. Sometimes, the variations can be substantial in quantitative or qualitative terms. The new approach is applied to choked-nozzle flow, isentropic flow, nonlinear wave propagation and flow across a shock wave, all for a real gas. Modifications are obtained for allowable mass flow through a choked nozzle, nozzle thrust, sonic wave speed, Riemann invariants, Prandtl’s shock relation and the Rankine–Hugoniot relations. Forced acoustic oscillations can show substantial augmentation of pressure amplitudes when real-gas effects are taken into account. Shocks at higher temperatures and pressures can have larger pressure jumps with real-gas effects. Weak shocks decay to zero strength at sonic speed. The proposed framework can rely on any cubic equation of state and can be applied to multicomponent flows or to more complex flow configurations.

Temporal instabilities of a planar liquid jet are studied using direct numerical simulation (DNS) of the incompressible Navier–Stokes equations with level-set (LS) and volume-of-fluid (VoF) surface tracking methods. $\unicode[STIX]{x1D706}_{2}$ contours are used to relate the vortex dynamics to the surface dynamics at different stages of the jet breakup – namely, lobe formation, lobe perforation, ligament formation, stretching and tearing. Three distinct breakup mechanisms are identified in the primary breakup, which are well categorized on the parameter space of gas Weber number ($We_{g}$) versus liquid Reynolds number ($Re_{l}$). These mechanisms are analysed here from a vortex dynamics perspective. Vortex dynamics explains the hairpin formation, and the interaction between the hairpins and the Kelvin–Helmholtz (KH) roller explains the perforation of the lobes, which is attributed to the streamwise overlapping of two oppositely oriented hairpin vortices on top and bottom of the lobe. The formation of corrugations on the lobe front edge at high $Re_{l}$ is also related to the location and structure of the hairpins with respect to the KH vortex. The lobe perforation and corrugation formation are inhibited at low $Re_{l}$ and low $We_{g}$ due to the high surface tension and viscous forces, which damp the small-scale corrugations and resist hole formation. Streamwise vorticity generation – resulting in three-dimensional instabilities – is mainly caused by vortex stretching and baroclinic torque at high and low density ratios, respectively. Generation of streamwise vortices and their interaction with spanwise vortices produce the liquid structures seen at various flow conditions. Understanding the liquid sheet breakup and the related vortex dynamics are crucial for controlling the droplet-size distribution in primary atomization.

This paper experimentally investigates the convection in a rapidly rotating tangent cylinder (TC), for Ekman numbers down to $E=3.36\times 10^{-6}$. The apparatus consists of a hemispherical fluid vessel heated in its centre by a protruding heating element of cylindrical shape. The resulting convection that develops above the heater, i.e. within the TC, is shown to set in for critical Rayleigh numbers and wavenumbers respectively scaling as $Ra_{c}\sim E^{-4/3}$ and $a_{c}\sim E^{-1/3}$ with the Ekman number $E$. Although exhibiting the same exponents as for plane rotating convection, these laws reflect much larger convective plumes at onset. The structure and dynamics of supercritical plumes are in fact closer to those found in solid rotating cylinders heated from below, suggesting that the confinement within the TC induced by the Taylor–Proudman constraint influences convection in a similar way as solid walls would do. There is a further similarity in that the critical modes in the TC all exhibit a slow retrograde precession at onset. In supercritical regimes, the precession evolves into a thermal wind with a complex structure featuring retrograde rotation at high latitude and either prograde or retrograde rotation at low latitude (close to the heater), depending on the criticality and the Ekman number. The intensity of the thermal wind measured by the Rossby number $Ro$ scales as $Ro\simeq 5.33(Ra_{q}^{\ast })^{0.51}$ with the Rayleigh number based on the heat flux $Ra_{q}^{\ast }\in [10^{-9},10^{-6}]$. This scaling is in agreement with heuristic predictions and previous experiments where the thermal wind is determined by the azimuthal curl of the balance between the Coriolis force and buoyancy. Within the range $Ra\in [2\times 10^{7},10^{9}]$ which we explored, we also observe a transition in the heat transfer through the TC from a diffusivity-free regime where $Nu\simeq 0.38E^{2}Ra^{1.58}$ to a rotation-independent regime where $Nu\simeq 0.2Ra^{0.33}$.

We present a generalisation of the Kármán–Howarth–Monin (K–H–M) equation to include variable-density (VD) effects. The derived equation (i) reduces to the original K–H–M equation when density is a constant and (ii) leads to a VD analogue of the $4/5$-law with the same value of constant ($=4/5$) appearing as the prefactor of the dissipation rate. The equation is employed to understand negative turbulent kinetic energy production in a $\text{SF}_{6}$ turbulent round jet with an initial density ratio of 4.2. From a Reynolds-averaged Navier–Stokes (RANS) perspective, negative production means that the mean flow is strengthened at the expense of the energy of turbulent fluctuations. We show that the associated energy transfer is accomplished by the deformation of smaller turbulent eddies into large ones in the development region of the jet and is captured by the linear scale-by-scale energy transfer term in the VD K–H–M equation. The nonlinear transfer term of the VD K–H–M equation depicts a conventional forward cascade for all eddies having a size less than the Eulerian integral length scale, regardless of their orientation. The net effect is a retarded energy cascade in the non-Boussinesq jet that has not been accounted for by existing turbulence theories. Implications of this observation for turbulence modelling are discussed.

High-fidelity simulations of turbulent flow through a channel with a rough wall, followed by a smooth wall, demonstrate a high degree of non-equilibrium within the recovery region. In fact, the recovery of all the flow statistics studied is incomplete by the streamwise exit of the computational domain. Above a thin wall layer, turbulence intensities significantly higher than fully developed, smooth-wall levels persist in the developing region. Within the thin wall layer, the profile shapes for turbulence stresses recover very quickly and wall-normal locations of characteristic peaks are established. However, even in this thin layer, complete recovery of magnitudes of turbulence stresses is exceptionally slow. A similar initially swift but eventually incomplete and slow relaxation behaviour is also shown by the skin friction. Between the turbulence shear and streamwise stresses, the turbulence shear stress shows a comparatively quick rate of recovery above a thin wall layer. Over the developing smooth wall, the balance is not merely between fluxes due to pressure and shear stresses. Strong momentum fluxes, which are directly influenced by the upstream roughness size, contribute significantly to this balance. Approximate curve fits estimate the streamwise distance required by the outer peaks of Reynolds stresses to attain near-fully-developed levels at approximately $20\unicode[STIX]{x1D6FF}{-}25\unicode[STIX]{x1D6FF}$, with $\unicode[STIX]{x1D6FF}$ being the channel half height. An even longer distance, of more than $50\unicode[STIX]{x1D6FF}$, might be needed by the mean velocity to approach near-fully-developed magnitudes. Visualizations and correlations show that large-scale eddies that are created above the roughness persist downstream, and sporadically perturb the elongated streaks. These streaks of alternating high and low momentum appear almost instantly after the roughness is removed. The mean flow does not re-establish an equilibrium log layer within the computational domain, and the velocity deficit created by the roughness continues throughout the domain. On the step change in roughness, near the wall, profiles for turbulence kinetic energy dissipation rate, $\unicode[STIX]{x1D716}$, and energy spectra indicate a sharp reduction in energy at small scales. Despite this, reversion towards equilibrium smooth-wall levels is slow, and ultimately incomplete, due to a rather slow adjustment of the turbulence cascade. The non-dimensional roughness height, $k^{+}$ ranges from 42 to 254 and the friction velocity Reynolds number at the smooth wall, $Re_{\unicode[STIX]{x1D70F}S}$, ranges from 284 to 1160 in the various simulations.

We use interface-resolved numerical simulations to study finite-size effects in turbulent channel flow of neutrally buoyant spheres. Two cases with particle sizes differing by a factor of two, at the same solid volume fraction of 20 % and bulk Reynolds number are considered. These are complemented with two reference single-phase flows: the unladen case, and the flow of a Newtonian fluid with the effective suspension viscosity of the same mixture in the laminar regime. As recently highlighted in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501), a particle–wall layer is responsible for deviations of the mesoscale-averaged statistics from what is observed in the continuum limit where the suspension is modelled as a Newtonian fluid with (higher) effective viscosity. Here we investigate in detail the fluid and particle dynamics inside this layer and in the bulk. In the particle–wall layer, the near-wall inhomogeneity has an influence on the suspension microstructure over a distance proportional to the particle size. In this layer, particles have a significant (apparent) slip velocity that is reflected in the distribution of wall shear stresses. This is characterized by extreme events (both much higher and much lower than the mean). Based on these observations we provide a scaling for the particle-to-fluid apparent slip velocity as a function of the flow parameters. We also extend the scaling laws in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501) to second-order Eulerian statistics in the homogeneous suspension region away from the wall. The results show that finite-size effects in the bulk of the channel become important for larger particles, while negligible for lower-order statistics and smaller particles. Finally, we study the particle dynamics along the wall-normal direction. Our results suggest that single-point dispersion is dominated by particle–turbulence (and not particle–particle) interactions, while differences in two-point dispersion and collisional dynamics are consistent with a picture of shear-driven interactions.

A diagnostic analysis is made for the ageostrophic secondary circulation, buoyancy flux and frontogenetic tendency (SCFT) in upper-ocean submesoscale fronts and dense filaments under the combined influences of boundary-layer turbulent mixing, surface wind stress and surface gravity waves. The analysis is based on a momentum-balance approximation that neglects ageostrophic acceleration, and the surface wave effects are represented with a wave-averaged asymptotic theory based on the time scale separation between wave and current evolution. The wave’s Stokes-drift velocity $\boldsymbol{u}_{st}$ induces SCFT effects that are dominant in strong swell with weak turbulent mixing, and they combine with Ekman and turbulent thermal wind influences in more general situations near wind–wave equilibrium. The complementary effect of the submesoscale currents on the waves is weak for longer waves near the wind–wave or swell spectrum peak, but it is strong for shorter waves.

A finite difference scheme is proposed to solve the problem of axisymmetric Taylor bubbles rising at a constant velocity in a tube. A method to remove singularities from the numerical scheme is presented, allowing accurate computation of the bubbles with the inclusion of both gravity and surface tension. This paper confirms the long-held belief that the solution space of the axisymmetric Taylor bubble for small surface tension is qualitatively similar to that of the plane Taylor bubble. Furthermore, evidence suggesting that the solution selection mechanism associated with plane bubbles also occurs in the axisymmetric case is presented.

The thermodynamic regime of a complex mixture depends on the composition, the pressure and the temperature; the spinodal locus separates the regime of thermodynamic instability from the remainder of the phase space. Since diffusion is one of the phenomena affecting the local chemical composition, the first focus is here on evaluating diffusion models in the context of high-pressure (high-$p$) multispecies mixing and combustion. It is shown that the diffusion model equations previously used to create two high-$p$ direct numerical simulation (DNS) databases can reproduce classical experimental observations of uphill diffusion in an accurate spatiotemporal manner, whereas the popular model which has a diagonal diffusion matrix and uses a velocity correction lacks spatiotemporal accuracy. Further, a mathematical formalism is used to compute the spinodal locus for mixtures for which either experimental data or previous computations from the literature are available, and it is shown that the agreement of the present calculations with that previously existing information is excellent. Using the spinodal-calculation mathematical formalism, the aforementioned DNS databases are then examined to determine the thermodynamic regime of the mixture at important stages of the simulations. In the first subset of the DNS databases that portrays mixing of five species under high-$p$ conditions, this stage is that of the transitional state representing the individual time station at which each simulation, having been initiated in a laminar state, transitions to a state having turbulent characteristics. In the second subset of the DNS databases that portrays high-$p$ turbulent combustion, this stage represents the individual time station at the peak $p$ achieved during the calculations. In both databases, the influence of the initial Reynolds number, the free-stream composition and the free-stream $p$ is studied. The results show that in all cases the mixture is in the single-phase regime. The present DNS databases have only five species, but it is shown that the methodology for computing the spinodal locus can be applied to very complex mixtures, with examples given for a twelve-species mixture and surrogate diesel fuels, thereby boding well for determining the thermodynamic regime of practical mixtures in high-$p$ turbulent flow simulations for engineering applications. According to these calculations, diesel-fuel surrogates are always in the single-phase regime at injection-conditions $p$ and temperatures existing in diesel-engine combustion chambers.

We study the destabilization of a round liquid jet by a fast annular gas stream. We measure the frequency of the shear instability waves for several geometries and air/water velocities. We then carry out a linear stability analysis, and show that there are three competing mechanisms for the destabilization: a convective instability, an absolute instability driven by surface tension and an absolute instability driven by confinement. We compare the predictions of this analysis with experimental results, and propose scaling laws for wave frequency in each regime. We finally introduce criteria to predict the boundaries between these three regimes.

Natural convection produced by a non-uniform internal heat source is studied numerically. Our investigation is limited to a two-dimensional enclosure with an aspect ratio equal to two. The energy source is Joule dissipation produced by an electric potential applied through two electrodes corresponding to a fraction of the vertical walls. The system of conservative equations of mass, momentum, energy and electric potential is solved assuming the Boussinesq approximation with a discontinuous Galerkin finite element method integrated over time. Three parameters are involved in the problem: the Rayleigh number $Ra$, the Prandtl number $Pr$ and the electrode length $L_{e}$ normalized by the enclosure height. The numerical method has been validated in a case where electrodes have the same length as the vertical walls, leading to a uniform source term. The threshold of convection is established above a critical Rayleigh number, $Ra_{cr}=1702$. Due to asymmetric boundary conditions on thermal field, the onset of convection is characterized by a transcritical bifurcation. Reduction of the size of the electrodes (from bottom up) leads to disappearance of the convection threshold. As soon as the electrode length is smaller than the cavity height, convection occurs even for small Rayleigh numbers below the critical value determined previously. At moderate Rayleigh number, the flow structure is mainly composed of a left clockwise rotation cell and a right anticlockwise rotation cell symmetrically spreading around the vertical middle axis of the enclosure. Numerical simulations have been performed for a specific $L_{e}=2/3$ with $Ra\in [1;10^{5}]$ and $Pr\in [1;10^{3}]$. Four kinds of flow solutions are established, characterized by a two-cell symmetric steady-state structure with down-flow in the middle of the cavity for the first one. A first instability occurs for which a critical Rayleigh number depends strongly on the Prandtl number when $Pr<3$. The flow structure becomes asymmetric with only one steady-state cell. A second instability occurs above a second critical Rayleigh number that is quasiconstant when $Pr>10$. The flow above the second critical Rayleigh number becomes periodic in time, showing that the onset of unsteadiness is similar to the Hopf bifurcation. When $Pr<3$, a fourth steady-state solution is established when the Rayleigh number is larger than the second critical value, characterized by a steady-state structure with up-flow in the middle of the cavity.

Fakir-like superhydrophobic surfaces, formed by doubly periodic arrays of thin pillars that sustain a lubricating gas layer, exhibit giant liquid-slip lengths that scale as $\unicode[STIX]{x1D719}^{-1/2}$ relative to the periodicity, $\unicode[STIX]{x1D719}$ being the solid fraction (Ybert et al., Phys. Fluids, vol. 19, 2007, 123601). Considering arbitrarily shaped pillars distributed over an arbitrary Bravais lattice, we employ matched asymptotic expansions to calculate the slip-length tensor in the limit $\unicode[STIX]{x1D719}\rightarrow 0$. The leading $O(\unicode[STIX]{x1D719}^{-1/2})$ slip length is determined from a local analysis of an ‘inner’ region close to a single pillar, in conjunction with a global force balance. This leading term, which is independent of the lattice geometry, is related to the drag due to pure translation of a flattened disk shaped like the pillar cross-section; its calculation is illustrated for the case of elliptical pillars. The $O(1)$ slip length is associated with the excess velocity induced about a given pillar by all the others. Since the field induced by each pillar corresponds on the ‘outer’ lattice scale to a Stokeslet whose strength is fixed by the shear rate, the $O(1)$ slip length depends upon the lattice geometry but is independent of the cross-sectional shape. Its calculation entails asymptotic evaluation of singular lattice sums. Our approximations are in excellent agreement with existing numerical computations for both circular and square pillars.