When an impermeable spherical particle is suspended in stably stratified ambient salinity gradient, the curving of the constant-density surfaces gives rise to non-conservative body forces which cannot be supported by pressure gradients. As a result, fluid motion is engendered. Considering small particles, we here address the Stokes-flow limit and assess the role of solute advection, characterised by the Rayleigh number. For a purely diffusive salt transport, the velocity field attains an orientation-dependent non-decaying profile at large distances from the sphere, in contradiction to the fundamental requirement of velocity attenuation. The weak advection limit is non-uniformly valid, with advection of the ambient salinity profile entering the dominant balance at large distances. This limit is accordingly resolved via use of matched asymptotic expansions, with the fluid domain conceptually decomposed into two regions, one on the particle scale and the other on the stratification scale. The stratification-scale problem is solved using Fourier transforms, with the solution shown to trivially match the diffusive fields on the particle scale. Simplification of the stratification-scale solution on the symmetry axis reveals that the velocity decays inversely with the eighth power of distance, while the excess concentration undergoes a transition from a diffusive inverse-square scaling to a faster decay, inversely with the sixth power of distance. In the comparable two-dimensional problem the non-conservative body force instigates a velocity field that diverges at large distances, thus exacerbating the weak advection conundrum. In this problem, use of matched asymptotic expansions is necessary to uniquely determine the flow on the obstacle scale.

We investigate the propagation of counter-rotating convection rolls in Rayleigh–Bénard convection initiated locally in a quiescent fluid layer under supercritical conditions. The velocity of the front separating quiescent fluid from the forming convection rolls, and the wavenumber of the convection rolls remaining behind the front, are explored. We numerically investigate fronts of forming convection rolls over five orders of magnitude of the reduced Rayleigh number, $\epsilon$, in two- and three-dimensional domains, for a broad range of boundary conditions, and for different front initiation approaches. In all cases, the front velocity increases as $\epsilon ^{1/2}$ with increasing $\epsilon$ for $\epsilon \lesssim 1$ in agreement with predictions using the amplitude equation. The amplitude equation description of the front velocity remains accurate for $\epsilon \lesssim 10$ except when the Prandtl number is large which yields a velocity that is faster than predicted for a fluid layer far from threshold. The wavenumber of the convection rolls increases linearly with $\epsilon$ in agreement with the wavenumber that maximises the growth rate of perturbations in the linear regime. Farther from onset, the wavenumber growth transitions to a reduced scaling of $\epsilon ^{1/4}$ in agreement with predictions using the Swift–Hohenberg equation in the large-$\epsilon$ limit. The scalings describing the wavenumber variation with $\epsilon$ are independent of the domain geometry, boundary conditions and front initiation method. However, the front-selected wavenumber at criticality does not equal the critical wavenumber of the bulk instability, in general, and depends significantly upon these details. We compare our results with experimental measurements where possible.

We consider convection in an internally heated system with a nonlinear equation of state (EOS) using linear stability analysis, direct numerical simulation (DNS) and weakly nonlinear analysis. Motivated by melt ponds on Arctic sea ice, we consider a fluid layer subject to heating from radiative absorption, an isothermal lower boundary and an imperfectly conducting top boundary. A modified Fourier spectral method is employed in the linear stability analysis to ensure numerical convergence. We perform two-dimensional DNS for a wide range of parameters, alongside representative three-dimensional cases. Both linear theory and the DNS show that the nonlinear EOS significantly influences the instability characteristics. We define a dimensionless parameter $\kappa _n$ to quantify the level of nonlinearity of the EOS, so that the Rayleigh number $ \textit{Ra}$ and $\kappa _n PrRa^2$ (where $ \textit{Pr}$ is the Prandtl number) act as a measure of the buoyancy-driven forcing caused by the linear and nonlinear parts of the EOS, respectively. With sufficiently large $ \textit{Ra}$, and increasing $|\kappa _n|$, the system undergoes a transition between different modal structures. There is bottom-up convection from the lower boundary with small $|\kappa _n|$, and shallow top-down convection from the top boundary with large $|\kappa _n|$. For intermediate $|\kappa _n|$ and increasing $ \textit{Ra}$, we find successive alternating windows of stability, instability, reemergence of stability and finally instability. A weakly nonlinear analysis indicates that the bifurcations are supercritical. The implications for mixing in Arctic melt ponds are discussed. The net heat flux across the atmosphere-pond interface varies depending on the flow regime.

We derive asymptotic models for the ultimate regimes in horizontal convection (HC) and pure internally heated convection (IHC), in analogy with our recent (2024) extension of the ultimate-regime model for Rayleigh–Bénard convection (RBC). To derive the corresponding models for HC and IHC, we combine turbulent boundary-layer relations with the exact dissipation balances for these two systems. For HC, the resulting scaling relations are consistent with the rigorous transport bound of Siggers et al. (2004 J. Fluid Mech. vol. 517, 55–70). For pure IHC, they are consistent with the exact HC–IHC balance analogy of Wang et al. (2021 Geophys. Res. Lett. vol. 48, e2020GL091198) and with the rigorous bounds on the convective-flux asymmetry in the equal-temperature-plates configuration (Arslan et al. 2021 J. Fluid Mech. vol. 919, A15). The main difference between RBC and HC/IHC is that, in the latter two cases, the global kinetic-energy balance does not contain the additional response factor (dimensionless convective heat flux in HC or inverse bulk temperature in IHC), whereas it does in RBC. As a consequence, for fixed Prandtl number ${\textit{Pr}}$, the ultimate-regime scaling exponent is $1/3$ for both HC and IHC, rather than $1/2$ as in RBC.

We investigate the competition between horizontal convection (HC) and Rayleigh–Bénard convection (RBC) in a fluid layer subject to a uniform destabilising buoyancy flux at the bottom and a horizontally varying buoyancy distribution at the top. The RBC forcing imposes negative horizontal mean vertical buoyancy gradients at the top and bottom of the fluid layer. But if the HC forcing is sufficiently strong then the volume-averaged vertical buoyancy gradient, $\langle b_z \rangle$, is positive i.e. opposite in sign to destabilising RBC buoyancy gradients at the boundaries. If $\langle b_z \rangle \gt 0$ we say that the layer has been ‘restratified’. Using scaling analysis based on power integrals together with two-dimensional direct numerical simulations at Rayleigh numbers up to $10^{10}$, we identify two cases: a neutral stratification state, in which HC first offsets RBC so that $\langle b_z \rangle = 0$, and a strong stratification regime, in which HC dominates and $\langle b_z \rangle$ is opposite in sign, and greater in magnitude, than the prescribed destabilising vertical buoyancy gradient at the layer boundaries. For the range of parameters explored in this study, we derive scaling laws for the onset of these regimes in terms of the horizontal and vertical flux Rayleigh numbers, $\mathit{Ra_H}$ and $\mathit{Ra_V}$, finding $\mathit{Ra}^{\mathit{N}}_{\mathit{H}} \sim \mathit{Ra_V}^{4/5}$ for the neutral state and $\mathit{Ra}^{\mathit{S}}_{\mathit{H}} \sim \mathit{Ra_V}$ for the onset of strong stratification. The results highlight the controlling role of the top boundary layer in setting the mean stratification and clarify the conditions under which HC suppresses RBC. These findings are relevant to geophysical environments such as subglacial lakes, and the oceans of snowball Earth and icy moons, where bottom heating and horizontal buoyancy variations jointly shape ocean stratification.

We investigate the influence of the Prandtl number ($\textit{Pr}$) on penetrative internally heated convection (IHC) in both non-rotating and rotating regimes using three-dimensional direct numerical simulations. By varying $\textit{Pr}$ between 0.1 and 100, we show that the global mean temperature $\overline {\langle T \rangle }$ is not very sensitive to $\textit{Pr}$, and is primarily controlled by the dynamics of the unstably stratified top boundary layer. In contrast, the Prandtl number dictates the behaviour of the lower, stably stratified region and affects the vertical convective heat flux $\overline {\langle wT \rangle }$. In the non-rotating case, low-$\textit{Pr}$ fluids exhibit a ‘symmetry recovery’ where turbulent stirring agitates the stable layer, whereas high-$\textit{Pr}$ fluids transition towards a ‘dead zone’ of suppressed fluctuations. Under rotation, we find that $\overline {\langle wT \rangle }$ is enhanced across all Prandtl numbers, though global cooling efficiency, measured by the reduction in $\overline {\langle T \rangle }$, is only improved for $\textit{Pr}\geqslant 1$ due to the emergence of Ekman pumping. These results demonstrate that while IHC shares some scaling similarities with Rayleigh–Bénard convection at the top boundary, the internal stratification creates a unique sensitivity to $\textit{Pr}$ that is critical for understanding heat transport in planetary and stellar interiors.

Fluid pumping in a horizontal slot using the pattern interaction effect has been analysed. This pumping is of interest as it operates without external energy sources beyond those required to create the necessary heating patterns. Activation of this effect involves a combination of fixed surface topography and adjustable heating patterns. The flow rate, including its direction, can be controlled by moving the heating pattern relative to the groove pattern. This analysis extends that of Abtahi & Floryan (2017 J. Fluid Mech. vol. 826, pp. 553–582), who considered only small-amplitude grooves in which the achieved flow rate is proportional to the groove amplitude. Grooves with arbitrary amplitudes spanning the slot were considered in the current analysis, and their most effective heights and distributions have been identified. A detailed analysis was conducted of groove and heating patterns described by a single Fourier mode applied to one or both plates bounding the slot. In all cases, the flow rate increased proportionally to the groove amplitude until an excessively large amplitude caused flow choking, and to the heating intensity until saturation was reached. The groove wavenumber of approximately 0.8 was found to be the most effective in the case of one groove plate, and 0.5–0.7 in the case of two groove plates. The flow rate decreases rapidly at both smaller and larger wavenumbers. The largest flow rate was achieved by placing grooves on both plates to form a wavy slot, with hot spots positioned halfway between the groove peaks and troughs.

This study investigates convection in a non-isothermal spherical Taylor–Couette flow (sTC) under the influence of the dielectrophoretic (DEP) force. The convective flow is driven by differential rotation of the inner and outer boundaries rotating with $\varOmega$ and $\Delta {\varOmega }$ in combination of an electric tension applied between both shells to induce thermo-electrohydrodynamic (TEHD) convection. To understand the interaction between DEP force-driven and rotation-driven mechanisms, we first analysed TEHD convection and non-isothermal sTC flow independently. For the TEHD case, we establish scaling relations for heat transport by expressing the Nusselt number, ${\textit{Nu}}$, as a function of the electric Rayleigh number, ${\textit{Ra}}_{{E}}$, and the kinetic energy density, $\tilde {E}_k$. These relations are evaluated against classical models of convection to assess consistency and deviations. A similar approach was applied to the non-isothermal sTC flow in the absence of the DEP force, where we identified axisymmetric and non-axisymmetric flow regimes which were classified by ${\textit{Nu}}$, $\tilde {E}_k$ and $\Delta {\varOmega }$, and developed corresponding scaling relations. When both mechanisms were active, ${\textit{Nu}}$ generally increased, however, the DEP force locally suppressed angular momentum transport, especially near the equator. This interplay revealed three distinct regimes: (A) DEP force-dominated TEHD convection, (C) rotation-dominated non-isothermal sTC flow and (B) a transitional regime with reduced heat transport. A decomposition of a derived inflow Nusselt number, ${\textit{Nu}}^q$, based on conductive and convective contributions, further elucidated the underlying heat transport mechanism.

Natural convection within a heated, inclined slot with wavy walls is investigated. The coupling between the heating and topography patterns determines the properties of the flow and the effectiveness of this interaction can vary significantly as the slot inclination is adjusted. The analysis is two-dimensional and is likely to be a useful model for the flow in a slot of large cross-stream aspect ratio. Typically, the flow topology consists of a combination of rolls and stream tubes that carry the fluid along the conduit. It is shown that a judicious choice of inclination angle and careful positioning of the grooves relative to the temperature pattern can yield a flow rate greater than that achievable within a smooth slot. There is an optimal inclination and phase difference between the groove and heating patterns for which the flow rate is the greatest. The most effective inclination angle is a function of the wavelength and amplitude of the grooves, the heating intensity and the fluid Prandtl number.

This experimental study aims to clarify how and when a weak centrifugal force affects the turbulent rotating thermal convection system. For the bulk flow, the weak centrifugal effect is significant on long-time averaged flow fields, contrasting sharply with its negligible effect on the instantaneous fields. As for the sidewall flow, it is found that properties of the boundary zonal flow are influenced by the weak centrifugal force appreciably. The onset Froude number $\textit{Fr}_c^{\textit{local}}$, signalling when the weak centrifugal effects start to set in, is found to scale with Rayleigh number $Ra$ as $\textit{Fr}_c^{\textit{local}}\sim Ra^{0.55}$ over approximately two and a half decades of $Ra$. The underlying mechanism for this robust scaling is captured by the mechanism of local force balance, which involves three unknown local scales. With the help of both a viscous–Archimedean–Coriolis argument and the experimental data, these local scales are successfully resolved to reveal a consistent result with this 0.55 scaling.

The Earth’s quasi-biennial oscillation (QBO) is a natural example of wave–mean flow interaction and corresponds to the alternating directions of winds in the equatorial stratosphere. It is due to internal gravity waves (IGWs) generated in the underlying convective troposphere. In stars, a similar situation is predicted to occur, with the interaction of a stably stratified radiative zone and a convective zone. In this context, we investigate the dynamics of this reversing mean flow by modelling a stably stratified envelope and a convectively unstable core in polar geometry. Here, the coupling between the two zones is achieved self-consistently, and IGWs generated through convection lead to the formation of a reversing azimuthal mean flow in the upper layer. We characterise the mean flow oscillations by their periods, velocity amplitudes and regularity. Despite a continuous broad spectrum of IGWs, our work shows good qualitative agreement with the monochromatic model of Plumb & McEwan (1978, J. Atmos. Sci. vol. 35, no. 10, pp. 1827–1839). While the latter was originally developed in the context of the Earth’s QBO, then our study could prove relevant for its stellar counterpart in massive stars, which host convective cores and radiative envelopes.

This study investigates the heat-flux enhancement of convection flows inside a fluid layer bounded from the top and bottom by two inhomogeneous porous layers. The porous matrix is made of solid materials with very high diffusivity. The numerical results reveal that, compared with the traditional convection system, the heat flux is greatly increased when the thickness of porous layer is large enough. At small Rayleigh numbers, the enhancement is the result of the increase in effective diffusivity in the fluid-saturated porous layers and the reduction in flow friction at the porous interface. For large Rayleigh numbers, the permeable motions across the interfaces generate strong convective flux, which greatly increases the total heat flux. For the latter parameter range, the exponent of the power-law scaling between the Nusselt number and the Rayleigh number exceeds 1/2, which is the value of the ultimate scaling. Our findings are not only of great potential in heat management in various industrial applications but also imply that, in many natural systems with imperfect boundaries, the global heat flux may be much stronger than the prediction by using a convection system with perfect boundaries.

In many electrochemical systems, variations in fluid density due to salinity gradients are unavoidable, leading to solutally driven Rayleigh–Bénard convection (RBC). In this study, we perform direct numerical simulations and theoretical analyses of two-dimensional solutal convection near perfectly cation-selective membranes by incorporating buoyancy and electrostatic forces into the Navier–Stokes and Poisson–Nernst–Planck equations. When electroconvection (EC) is negligible, we observe a flow reversal of large-scale circulation (LSC) in salt-driven RBC within a square-cavity electrochemical system, triggered by the periodic reconfiguration of corner vortices. Furthermore, we found that the competition between RBC and EC determines the dominant flow pattern. The buoyancy-driven convection and the LSC are suppressed at sufficiently strong EC flow, leading to a transition from buoyancy-driven flow to electrically driven flow. Consequently, the flow structures into a pair of EC vortices, driven by strong electric field forces within the extended space charge layer. Using Grossmann–Lohse theory, we derive a critical scaling law that describes the flow pattern selection, governed by the combined effects of the Rayleigh number, voltage difference and hydrodynamic coupling coefficient. Our work presents a novel approach to controlling flow patterns, distinct from existing strategies in thermally driven RBC.

This work presents an analytical solution for the steady laminar wake generated by a finite wall segment acting as a sink for heat or mass transfer. The classical Lévêque solution is extended to include the wake region downstream of the active surface by employing Laplace transform methods to couple Dirichlet and Neumann boundary value problems through convolution identities. This yields a unified closed-form expression for the scalar field that reduces to the Lévêque result above the sink and provides a new analytical expression for the wake region. Numerical simulations confirm the analytical solution, with errors decreasing systematically under mesh refinement. The derived expressions enable direct calculation of scalar recovery at any point in the wake, providing essential information for designing segmented systems where wake interference between adjacent active elements must be predicted. The solution also serves as a benchmark for numerical methods solving mixed boundary value problems in convective transport.

We present a study on the melting dynamics of neighbouring ice bodies by means of idealised simulations, focusing on collective effects, with the goal of obtaining fundamental insight into how collective interactions influence the melting of ice. Two neighbouring (vertically or horizontally aligned), square-shaped and equally sized ice objects (size of the order of centimetres) are immersed in quiescent fresh water at a temperature of ${20}\,^\circ \textrm {C}$. By performing two-dimensional direct numerical simulations, and using the phase-field method to model the phase change, the collective melting of these objects is studied. When the objects are horizontally aligned, no significant influence of the neighbouring object on the melting time is observed. On the other hand, when vertically aligned, although the melting of the upper object is mostly unaffected, the melting time and the morphology of the lower ice body strongly depends on the initial inter-object distance. We report that the melting of the bottom object can be enhanced by more than 10 %, or delayed more than 20 %, displaying a non-monotonic dependence on the initial object size. We show that this behaviour results from a non-trivial competition between layering of cold fluid, which lowers the heat transfer, and convective flows, which favour mixing and heat transfer. For this melting in mixed convection, we were able to collapse our data onto a single curve.

We study the combined effects of natural convection and rotation on the dissolution of a solute in a solvent-filled circular cylinder. The density of the fluid increases with increasing concentration of the dissolved solute, and we model this using the Oberbeck–Boussinesq approximation. The underlying moving-boundary problem has been modelled by combining the Navier–Stokes equations with the advection–diffusion equation and a Stefan condition for the evolving solute–fluid interface. We use highly resolved numerical simulations to investigate the flow regimes, dissolution rates and mixing of the dissolved solute for $Sc = 1$, $Ra \in [10^5, 10^8]$ and $\varOmega \in [0, 2.5]$. In the absence of rotation and buoyancy, the distance of the interface from its initial position follows a square root relationship with time ($r_d \propto \sqrt {t}$), which ceases to exist at a later time due to the finite-size effect of the liquid domain. We then explore the rotation parameter, considering a range of rotation frequency – from smaller to larger, relative to the inverse of the buoyancy-induced time scale – and Rayleigh number. We show that the area of the dissolved solute varies nonlinearly with time depending on $Ra$ and $\varOmega$. The symmetry breaking of the interface is best described in terms of $Ra/\varOmega ^2$.

We present a framework to calculate the scale-resolved turbulent Prandtl number ${\textit{Pr}}_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh–Bénard mesoscale convection layer at a molecular Prandtl number of ${\textit{Pr}}=10^{-3}$. It builds on Kolmogorov’s refined similarity hypothesis of homogeneous isotropic fluid and passive scalar turbulence, based on log–normally distributed amplitudes of kinetic energy and scalar dissipation rates that are coarse-grained over variable scales $r$ in the inertial subrange. Our definitions of turbulent (or eddy) viscosity and diffusivity do not rely on mean gradient-based Boussinesq closures of Reynolds stresses and convective heat fluxes. Such gradients are practically absent or indefinite in the bulk. The present study is based on direct numerical simulation of plane-layer convection at an aspect ratio of $\varGamma =25$ for Rayleigh numbers $10^5\leqslant Ra\leqslant 10^7$. We find that the turbulent Prandtl number is effectively up to four orders of magnitude larger than the molecular one, ${\textit{Pr}}_t\sim 10$. This holds particularly for the upper end of the inertial subrange, where the eddy diffusivity exceeds the molecular value, $\kappa _e(r)\gt \kappa$. Highly inertial low-Prandtl-number convection becomes effectively a higher-Prandtl-number turbulent flow, when turbulent mixing processes on scales that reach into the inertial range are included. This might have some relevance for prominent low-Prandtl-number applications, such as solar convection.

We present a novel approach to harness the oscillation energy from cilia in chaotic flow to enhance scalar transport, addressing limitations of the laminar boundary layer. In contrast to the scallop theorem, where reciprocal motion yields negligible transport, coordinated rigid cilium oscillations in chaotic flow trigger boundary-layer resonance, significantly boosting scalar transport at specific frequencies. Under relatively high rigidity, the cilia undergo only small elastic deformations at the driving frequency, and their strokes remain nearly time symmetric. Nevertheless, unlike the classical expectation that reciprocal motion yields negligible transport, coordinated rigid cilium oscillations in chaotic flow trigger boundary-layer resonance, producing a sharp, frequency-selective boost in transport. At low to medium frequencies, cilium-driven fluid displacement enhances transport via vertical mixing. Above a critical frequency, rapid cilium motion induces unstable shear flow, generating coherent vortical structures that amplify mixing in chaotic flow regimes. These vortices, which interact with the inherent coherent structures of the chaotic flow, dramatically improve the efficiency of transport. Our findings reveal a dynamic coupling between cilium-driven resonance and chaotic flow coherent structures, providing a paradigm for optimising transport in thermal systems through active flow control.