We studied the transport and deposition behaviour of point particles in Rayleigh–Bénard convection cells subjected to Couette-type wall shear. Direct numerical simulations (DNSs) are performed for Rayleigh number ($Ra$) in the range $10^{7} \leq Ra \leq ~10^9$ with a fixed Prandtl number $Pr = 0.71$, while the wall-shear Reynolds number ($Re_w$) is in the range $0 \leq Re_w \leq ~12\,000$. With the increase of $Re_w$, the large-scale rolls expanded horizontally, evolving into zonal flow in two-dimensional simulations or streamwise-oriented rolls in three-dimensional simulations. We observed that, for particles with a small Stokes number ($St$), they either circulated within the large-scale rolls when buoyancy dominated or drifted near the walls when shear dominated. For medium $St$ particles, pronounced spatial inhomogeneity and preferential concentration were observed regardless of the prevailing flow state. For large $St$ particles, the turbulent flow structure had a minor influence on the particles’ motion; although clustering still occurred, wall shear had a negligible influence compared with that for medium $St$ particles. We then presented the settling curves to quantify the particle deposition ratio on the walls. Our DNS results aligned well with previous theoretical predictions, which state that small $St$ particles settle with an exponential deposition ratio and large $St$ particles settle with a linear deposition ratio. For medium $St$ particles, where complex particle–turbulence interaction emerges, we developed a new model describing the settling process with an initial linear stage followed by a nonlinear stage. Unknown parameters in our model can be determined either by fitting the settling curves or using empirical relations. Compared with DNS results, our model also accurately predicts the average residence time across a wide range of $St$ for various $Re_w$.

We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with vertical and horizontal temperature gradients. In particular, the interaction of rotation with the horizontal temperature gradient drives a ‘thermal wind’ shear flow whose strength is controlled by the horizontal temperature gradient. Varying this parameter therefore systematically alters the ordering of the shearing time scale, the convective time scale and the correlation time scale. We demonstrate that quasilinear-type approximations work well when the shearing time scale or the correlation time scale is sufficiently short. In all cases, the generalised quasilinear approximation systematically outperforms the quasilinear approximation. We discuss the consequences for statistical theories of turbulence interacting with mean gradients.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number $Gr$, the Prandtl number $Pr$, and the Reynolds number $Re$ of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number $Ri=Gr/Re^2$. We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher $Gr$, and a drop in the heat flux relative to natural convection for $Ri=O(1)$. This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

We study the effects of Prandtl number $\mathit {Pr}$ and Rayleigh number $\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range $10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$, the viscous dissipation scales as $\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$, which is based on the observation that enstrophy $\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$, and the Nusselt number tends to follow the ‘ultimate’ scaling $\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of $\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum $\hat {E}_u(k) \propto k^{-2.3}$, which is close to $k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum $\hat {E}_\theta (k) \propto k^{-1.2}$, which deviates from the expected $k^{-7/5}$. Finally, at $\mathit {Pr} \to 0$ and $\infty$, we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.

We consider the two-dimensional Rayleigh–Bénard convection problem between Navier-slip fixed-temperature boundary conditions, and present a new upper bound for the Nusselt number (${\textit {Nu}}$). The result, based on a localization principle for the Nusselt number and an interpolation bound, exploits the regularity of the flow. On one hand our method yields a shorter proof of the celebrated result of Whitehead & Doering (Phys. Rev. Lett., vol. 106, 2011, 244501) in the case of free-slip boundary conditions. On the other hand, its combination with a new, refined estimate for the pressure gives a substantial improvement of the interpolation bounds in Drivas et al. (Phil. Trans. R. Soc. A, vol. 380, issue 2225, 2022, 20210025) for slippery boundaries. A rich description of the scaling behaviour arises from our result: depending on the magnitude of the Prandtl number (${\textit {Pr}}$) and slip length, our upper bounds indicate five possible scaling laws (where ${\textit {Ra}}$ is the Rayleigh number): ${\textit {Nu}}\sim (L_s^{-1}\,{\textit {Ra}})^{{1}/{3}}$, ${\textit {Nu}}\sim (L_s^{-2/5}\,{\textit {Ra}})^{{5}/{13}}$, ${\textit {Nu}}\sim {\textit {Ra}}^{{5}/{12}}$, ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-4/3}\,{\textit {Ra}})^{{1}/{2}}$ and ${\textit {Nu}}\sim {\textit {Pr}}^{-1/6}(L_s^{-1/3}\,{\textit {Ra}})^{{1}/{2}}$.

We experimentally and numerically characterize rapidly rotating radiatively driven thermal convection, beyond the sole heat transport measurements reported by Bouillaut et al. (Proc. Natl Acad. Sci., vol. 118, 2021, e2105015118). Based on a suite of direct numerical simulations (DNS) and additional processing of the experimental data collected by Bouillaut et al. (Proc. Natl Acad. Sci., vol. 118, 2021, e2105015118), we report the simultaneous validation of the scaling predictions of the ‘geostrophic turbulence’ regime – the diffusivity-free or ‘ultimate’ regime of rapidly rotating convection – for the heat transport and the temperature fluctuations. Following such cross-validation between DNS and laboratory experiments, we further process the numerical data to validate the ‘geostrophic turbulence’ scaling predictions for the flow velocity and horizontal scale. Radiatively driven convection thus appears as a versatile set-up for the laboratory observation of the diffusivity-free regimes of various convective flows of geophysical and/or astrophysical interest.

We report on an experimental study of turbulent Rayleigh–Bénard convection with asymmetric top and bottom plates. The plates are covered with pyramid-shaped roughness elements whose aspect ratios are $\lambda =1$ or $\lambda =4$. In the low-Rayleigh-number regime ($Ra<1.9\times 10^9$), the heat transport efficiencies in the asymmetric cells, characterized by the Nusselt number, are smaller than those measured in a symmetric $\lambda = 1$ cell and are greater than those for a symmetric $\lambda = 4$ cell, whereas in the high-Rayleigh-number regime ($Ra>1.9\times 10^9$), the Nusselt numbers of the asymmetric cells are, in turn, greater than those for the symmetric cell with $\lambda = 1$ and smaller than those for the symmetric cell with $\lambda = 4$. In addition, the heat transports of individual plates are studied based on the temperature drops across both halves of the cell. In the low-$Ra$ regime, the $\lambda =1$ plate shows higher heat transfer than the $\lambda =4$ plate, while for the high-$Ra$ regime, the $\lambda =4$ plate shows a higher heat transport ability. In both regimes, the individual Nusselt number of the plate with lower heat transfer is insensitive to the topology of the other plate. Besides, it is found that the symmetry of the centre temperature distribution is robust to the symmetry breaking of boundary topographies. For the $Ra$ range explored, a weak temperature inversion is observed in the bulk of asymmetric rough cells. Finally, we remark that the temperature fluctuation at the cell centre and the Reynolds number associated with the large-scale circulation show universal power laws in terms of the flux Rayleigh number as $\sigma _{T_{c}}\sim Ra_F^{0.68}$ and $Re_{LSC}\sim Ra_F^{0.36}$, respectively.

We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

We consider an internally heated fluid between parallel plates with fixed thermal fluxes. For a large class of heat sources that vary in the direction of gravity, we prove that $\smash { \smash {{\langle {\delta T} \rangle _h}} } \geq \sigma R^{-1/3} - \mu$, where $\smash { \smash {{\langle {\delta T} \rangle _h}} }$ is the average temperature difference between the bottom and top plates, $R$ is a ‘flux’ Rayleigh number and the constants $\sigma,\mu >0$ depend on the geometric properties of the internal heating. This result implies that mean downward conduction (for which $\smash { \smash {{\langle {\delta T} \rangle _h}} }< 0$) is impossible for a range of Rayleigh numbers smaller than a critical value $R_0:=(\sigma /\mu )^{3}$. The bound demonstrates that $R_0$ depends on the heating distribution and can be made arbitrarily large by concentrating the heating near the bottom plate. However, for any given fixed heating profile of the class we consider, the corresponding value of $R_0$ is always finite. This points to a fundamental difference between internally heated convection and its limiting case of Rayleigh–Bénard convection with fixed-flux boundary conditions, for which $\smash {{\langle {\delta T} \rangle _h}}$ is known to be positive for all $R$.

We perform linear stability analysis and direct numerical simulations to study the effect of the radius ratio on the instability and flow characteristics of the sheared annular centrifugal Rayleigh–Bénard convection, where the cold inner cylinder and the hot outer cylinder rotate with a small angular velocity difference. With the shear enhancement, the thermal convection is suppressed and finally becomes stable for different radius ratios $\{\eta \in \mathbb {R}|0.2\leqslant \eta \le 0.95\}$. Considering the inhomogeneous distribution of shear stresses in the base flow, a new global Richardson number $Ri_g$ is defined and the marginal-state curves for different radius ratios are successfully unified in the parameter domain of $Ri_g$ and the Rayleigh number $Ra$. The results are consistent with the marginal-state curve of the wall-sheared classical Rayleigh–Bénard convection in the streamwise direction, demonstrating that the basic stabilization mechanisms are identical. Moreover, systems with small radius ratios exhibit greater geometric asymmetry. On the one hand, this results in a smaller equivalent aspect ratio for the system, accommodating fewer convection roll pairs; fewer roll pairs are more likely to cause a transition in the flow structure during shear enhancement. On the other hand, the shear distribution is more inhomogeneous, allowing for an outward shift of the convection region and the elevation of bulk temperature under strong shear.

This study proposes a new mechanism that can lead to layering or convection from the finite amplitude perturbation acting on the double diffusive convection with uniform background shear. We focus on the double diffusive convection in the diffusive regime with the cold fresh water laying above the warm salty water. We demonstrate that, although the unperturbed system is linearly stable, the finite amplitude perturbation can trigger the initial flow motions which subsequently obtain energy from the gravitational potential energy and from the uniform background shear, and evolve to layering or convection. By using the linear stability analysis for the initial growth stage and the energy analysis for the following transitional stage, the critical Richardson number can be predicted theoretically. Here the Richardson number measures the relative strength of stratification to the background shear. The dominant wavenumbers and the growth rates of the corresponding modes given by linear theory agree well with the two-dimensional direct numerical simulations, and so does the critical Richardson number predicted by the theoretical model. The layering state is dominated by the double diffusion process, while the convection state at smaller Richardson number exhibits stronger influences from shear and generates smaller heat and salinity fluxes. The theoretical model is further applied to the parameter range which is relevant to the real oceanic environments and reveals that for the typical density ratio observed in the staircase regions in the Arctic Ocean, the current mechanism can lead to layering for relatively weak shear.

We present a systematic study on the effects of small aspect ratios $\varGamma$ on heat transport in liquid metal convection with a Prandtl number of $Pr=0.029$. The study covers $1/20\le \varGamma \le 1$ experimentally and $1/50\le \varGamma \le 1$ numerically, and a Rayleigh number $Ra$ range of $4\times 10^3 \le Ra \le 7\times 10^{9}$. It is found experimentally that the local effective heat transport scaling exponent $\gamma$ changes with both $Ra$ and $\varGamma$, attaining a $\varGamma$-dependent maximum value before transition-to-turbulence and approaches $\gamma =0.25$ in the turbulence state as $Ra$ increases. Just above the onset of convection, Shishkina (Phys. Rev. Fluids, vol 6, 2021, 090502) derived a length scale $\ell =H/(1+1.49\varGamma ^{-2})^{1/3}$. Our numerical study shows $Ra_{\ell }$, i.e. $Ra$ based on $\ell$, serves as a proper control parameter for heat transport above the onset with $Nu-1=0.018(1+0.34/\varGamma ^2)(Ra/Ra_{c,\varGamma }-1)$. Here $Ra_{c,\varGamma }$ represents the $\varGamma$-dependent critical $Ra$ for the onset of convection and $Nu$ is the Nusselt number. In the turbulent state, for a general scaling law of $Nu-1\sim Ra^\alpha$, we propose a length scale $\ell = H/(1+1.49\varGamma ^{-2})^{1/[3(1-\alpha )]}$. In the case of turbulent liquid metal convection with $\alpha =1/4$, our measurement shows that the heat transport will become weakly dependent on $\varGamma$ with $Ra_{\ell }\equiv Ra/(1+1.49\varGamma ^{-2})^{4/3} \ge 7\times 10^5$. Finally, once the flow becomes time-dependent, the growth rate of $Nu$ with $Ra$ declines compared with the linear growth rate in the convection state. A hysteresis is observed in a $\varGamma =1/3$ cell when the flow becomes time-dependent. Measurements of the large-scale circulation suggest the hysteresis is caused by the system switching from a single-roll-mode to a double-roll-mode in an oscillation state.

To date, a comprehensive understanding of the influence of the Prandtl number ($Pr$) on flow topology in turbulent Rayleigh–Bénard convection (RBC) remains elusive. In this study, we present an experimental investigation into the evolution of flow topology in quasi-two-dimensional turbulent RBC with $7.0 \leq Pr \leq 244.2$ and $2.03\times 10^{8} \leq Ra \leq 2.81\times 10^{9}$. Particle image velocimetry (PIV) measurements reveal the flow transitions from multiple-roll state to single-roll state with increasing $Ra$, and the transition is hindered with increasing $Pr$, i.e. the transitional Rayleigh number $Ra_t$ increases with $Pr$. We mapped out a phase diagram on the flow topology change on $Ra$ and $Pr$, and identified the scaling of $Ra_t$ on $Pr$: $Ra_t \sim Pr^{0.93}$ in the low $Pr$ range, and $Ra_t \sim Pr^{3.3}$ in the high $Pr$ range. The scaling in the low $Pr$ range is consistent with the model of balance of energy dissipation time and plume travel time that we proposed in our previous study, while the scaling in the high $Pr$ range implies a new governing mechanism. For the first time, the scaling of $Re$ on $Ra$ and $Pr$ is acquired through full-field PIV velocity measurement, $Re \sim Ra^{0.63}\,Pr^{-0.87}$. We also propose that increasing horizontal velocity promotes the formation of the large-scale circulation (LSC), especially for the high $Pr$ case. Our proposal was verified by achieving LSC through introducing horizontal driving force $Ra_H$ by tilting the convection cell with a small angle.

We present a theory to describe the Nusselt number, $\operatorname {\mathit {Nu}}$, corresponding to the heat or mass flux, as a function of the Rayleigh–Darcy number, $\operatorname {\mathit {Ra}}$, the ratio of buoyant driving force over diffusive dissipation, in convective porous media flows. First, we derive exact relationships within the system for the kinetic energy and the thermal dissipation rate. Second, by segregating the thermal dissipation rate into contributions from the boundary layer and the bulk, which is inspired by the ideas of the Grossmann and Lohse theory (J. Fluid Mech., vol. 407, 2000; Phys. Rev. Lett., vol. 86, 2001), we derive the scaling relation for $\operatorname {\mathit {Nu}}$ as a function of $\operatorname {\mathit {Ra}}$ and provide a robust theoretical explanation for the empirical relations proposed in previous studies. Specifically, by incorporating the length scale of the flow structure into the theory, we demonstrate why heat or mass transport differs between two-dimensional and three-dimensional porous media convection. Our model is in excellent agreement with the data obtained from numerical simulations, affirming its validity and predictive capabilities.

Three-dimensional direct numerical simulations of rotating Rayleigh–Bénard convection in the planar geometry with no-slip top and bottom and periodic lateral boundary conditions are performed for a broad parameter range with the Rayleigh number spanning in $5\times 10^{6}\leq Ra \leq 5\times 10^{13}$, Ekman number within $5\times 10^{-9}\leq Ek \leq 5\times 10^{-5}$ and Prandtl number $Pr=1$. The thermal and Ekman boundary layer (BL) statistics, temperature drop within the thermal BL, interior temperature gradient and scaling behaviours of the heat and momentum transports (reflected in the Nusselt $Nu$ and Reynolds numbers $Re$) as well as the convective length scale are investigated across various flow regimes. The global and local momentum transports are examined via the $Re$ scaling derived from the classical theoretical balances of viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA) forces. The VAC-based $Re$ scaling is shown to agree well with the data in the cellular and columnar regimes, where the characteristic convective length scales as the onset length scale ${\sim } Ek^{1/3}$, while the CIA-based $Re$ scaling and the inertia length scale $\sim (ReEk)^{1/2}$ work well in the geostrophic turbulence regime for $Ek\leq 1.5\times 10^{-8}$. The examinations of $Nu$, global and local $Re$, and convective length scale as well as the temperature drop within the thermal BL and its thickness scaling behaviours, indicate that for extreme parameters of $Ek\leq 1.5\times 10^{-8}$ and $80\lesssim RaEk^{4/3}\lesssim 200$, we have reached the diffusion-free geostrophic turbulence regime.

The validity of the Oberbeck–Boussinesq (OB) approximation in Rayleigh–Bénard (RB) convection is studied using the Gray & Giorgini (Intl J. Heat Mass Transfer, vol. 19, 1976, pp. 545–551) criterion that requires that the residuals, i.e. the terms that distinguish the full governing equations from their OB approximations, are kept below a certain small threshold $\hat {\sigma }$. This gives constraints on the temperature and pressure variations of the fluid properties (density, absolute viscosity, specific heat at constant pressure $c_p$, thermal expansion coefficient and thermal conductivity) and on the magnitudes of the pressure work and viscous dissipation terms in the heat equation, which all can be formulated as bounds regarding the maximum temperature difference in the system, $\varDelta$, and the container height, $L$. Thus for any given fluid and $\hat {\sigma }$, one can calculate the OB-validity region (in terms of $\varDelta$ and $L$) and also the maximum achievable Rayleigh number ${{Ra}}_{max,\hat {\sigma }}$, and we did so for fluids water, air, helium and pressurized SF$_6$ at room temperature, and cryogenic helium, for $\hat {\sigma }=5\,\%$, $10\,\%$ and $20\,\%$. For the most popular fluids in high-${{Ra}}$ RB measurements, which are cryogenic helium and pressurized SF$_6$, we have identified the most critical residual, which is associated with the temperature dependence of $c_p$. Our direct numerical simulations (DNS) showed, however, that even when the values of $c_p$ can differ almost twice within the convection cell, this feature alone cannot explain a sudden and strong enhancement in the heat transport in the system, compared with its OB analogue.

The geostrophic turbulence in rapidly rotating thermal convection exhibits characteristics shared by many highly turbulent geophysical and astrophysical flows. In this regime, the convective length and velocity scales and heat flux are all diffusion-free, i.e. independent of the viscosity and thermal diffusivity. Our direct numerical simulations (DNS) of rotating Rayleigh–Bénard convection in domains with no-slip top and bottom and periodic lateral boundary conditions for a fluid with the Prandtl number $Pr=1$ and extreme buoyancy and rotation parameters (the Rayleigh number up to $Ra=3\times 10^{13}$ and the Ekman number down to $Ek=5\times 10^{-9}$) indeed demonstrate all these diffusion-free scaling relations, in particular, that the dimensionless convective heat transport scales with the supercriticality parameter $\widetilde {Ra}\equiv Ra\, Ek^{4/3}$ as $Nu-1\propto \widetilde {Ra}^{3/2}$, where $Nu$ is the Nusselt number. We further derive and verify in the DNS that with the decreasing $\widetilde {Ra}$, the geostrophic turbulence regime undergoes a transition into another geostrophic regime, the convective heat transport in this regime is characterized by a very steep $\widetilde {Ra}$-dependence, $Nu-1\propto \widetilde {Ra}^{3}$.

We report a home-built velocity-gradient-tensor-resolved particle image velocimetry (VGTR-PIV) system which spatio-temporally resolves all components of the velocity gradient tensor. This technique is applied to the paradigmatic turbulent Rayleigh–Bénard convection system in a cylindrical cell at three representative positions, i.e. centre, side and bottom regions. The VGTR-PIV system allows us to directly measure, for the first time, the spatio-temporally resolved energy dissipation rate and enstrophy in turbulent thermal convection. In the experiment, the Rayleigh number $Ra$ varied in the range $2 \times 10^8 \leqslant Ra \leqslant 8 \times 10^9$ and the Prandtl number $Pr$ was fixed at $Pr = 4.34$. Compared with the fully resolved energy dissipation rate $\varepsilon$, the pseudo-dissipation provides the best estimate within $3\,\%$, the planar (two-dimensional) surrogate has a larger relative error and the one-dimensional surrogate leads to the largest error. The power-law scalings of the time-averaged energy dissipation rate with the Rayleigh number follow $\langle \varepsilon _c \rangle _t / (\nu ^3 H^{-4}) = 9.86 \times 10^{-6} Ra^{1.54 \pm 0.02}$, $\langle \varepsilon _s \rangle _t / (\nu ^3 H^{-4}) = 9.26 \times 10^{-3} Ra^{1.25 \pm 0.02}$ and $\langle \varepsilon _b \rangle _t / (\nu ^3 H^{-4}) = 2.70 \times 10^{-2} Ra^{1.23 \pm 0.02}$ in the centre, side and bottom regions, respectively where $\nu$ is dynamic viscosity and $H$ is cell height. These scaling relations, along with our earlier measured time-averaged energy dissipation rate at the bottom wall surface $\langle \varepsilon _w \rangle _t / (\nu ^3 H^{-4}) = 9.65 \times 10^{-2} Ra^{1.25 \pm 0.02}$ (J. Fluid Mech., vol. 947, 2022, A15), provide important constraints against which theoretical models may be tested. For the centre and side locations in the convection cell, the probability density functions (p.d.f.s) of the energy dissipation rate and enstrophy both follow a stretched exponential distribution. For the bottom region, the p.d.f.s of dissipation and enstrophy exhibit a stretched exponential distribution outside the viscous boundary layer and an exponential distribution inside the viscous boundary layer. It is also found that extreme events with high dissipation are the most intermittent in the side region, whereas the bottom region is less intermittent than the cell centre.

The present work focuses on the effect of rough horizontal boundaries on the heat transfer in rotating Rayleigh–Bénard convection. We measure the non-dimensional heat transfer, the Nusselt number $Nu$, for various strengths of the buoyancy forcing characterized by the Rayleigh number $Ra$ (${10^5}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }Ra\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }5 \times {10^8}$), and rotation rates characterized by the Ekman number E ($1.4 \times {10^{ - 5}}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }E\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }7.6 \times {10^{ - 4}}$) for aspect ratios $\varGamma \approx 1$, $2.8$ and $6.7$. Similar to rotating convection with smooth horizontal boundaries, the so-called rotationally constrained (RC), rotation-affected (RA) and rotation-unaffected (RuA) regimes of heat transfer seem to persist for rough horizontal boundaries. However, the transition from the RC regime to RA regime occurs at a lower Rayleigh number for rough boundaries. For all experiments with rough boundaries in this study, the thermal and Ekman boundary layers are in a perturbed state, leading to a significant enhancement in the heat transfer as compared with that for smooth walls. However, the enhancement in heat transfer due to wall roughness is observed to attain a maximum in the RC regime. We perform companion direct numerical simulations of rotating convection over smooth walls to suggest a phenomenology explaining this observation. We propose that the heat transfer enhancement due to wall roughness reaches a maximum when the strength and coherence of the columnar structures are both significant, which enables efficient vertical transport of the additional thermal anomalies generated by the roughness at the top and bottom walls.