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, $Nu-1\propto \widetilde {Ra}^{3}$.

In magnetoconvection, the flow of an electromagnetically conductive fluid is driven by a combination of buoyancy forces, which create the fluid motion due to thermal expansion and contraction, and Lorentz forces, which distort the convective flow structure in the presence of a magnetic field. The differences in the global flow structures in the buoyancy-dominated and Lorentz-force-dominated regimes lead to different heat transport properties in these regimes, reflected in distinct dimensionless scaling relations of the global heat flux (Nusselt number $Nu$) versus the strength of buoyancy (Rayleigh number $Ra$) and electromagnetic forces (Hartmann number $Ha$). Here, we propose a theoretical model for the transition between these two regimes for the case of a static vertical magnetic field applied across a convective fluid layer confined between two isothermal, a lower warmer and an upper colder, horizontal surfaces. The model suggests that the scaling exponents $\gamma$ in the buoyancy-dominated regime, $Nu\sim Ra ^\gamma$, and $\xi$ in the Lorentz-force-dominated regime, $Nu\sim (Ha^{-2}Ra)^\xi$, are related as $\xi =\gamma /(1-2\gamma )$, and the onset of the transition scales with $Ha^{-1/\gamma }Ra$. These theoretical results are supported by our direct numerical simulations for $10\leq Ha\leq 2000$, Prandtl number $Pr=0.025$ and $Ra$ up to $10^9$ and data from the literature.

In Rayleigh–Bénard convection, the size of a flow domain and its aspect ratio $\varGamma$ (a ratio between the spatial length and height of the domain) affect the shape of the large-scale circulation. For some aspect ratios, the flow dynamics includes a three-dimensional oscillatory mode known as a jump rope vortex (JRV); however, the effects of varying aspect ratios on this mode are not well investigated. In this paper, we study these aspect ratio effects in liquid metals, for a low Prandtl number ${{Pr}}=0.03$. Direct numerical simulations and experiments are carried out for a Rayleigh number range $2.9 \times 10^4 \leq {{Ra}} \leq 1.6 \times 10^6$ and square cuboid domains with $\varGamma =2$, $2.5$, $3$ and $5$. Our study demonstrates that a repeating pattern of a JRV encountered at aspect ratio $\varGamma \approx 2.5$ is the basic structural unit that builds up to a lattice of interlaced JRVs at the largest aspect ratio. The size of the domain determines how many structural units are self-organised within the domain; the number of the realised units is expected to scale as $\varGamma ^2$ with sufficiently large and growing $\varGamma$. We find the oscillatory modes for all investigated $\varGamma$; however, they are more pronounced for $\varGamma =2.5$ and $5$. Future studies for large-aspect-ratio domains of different shapes would enhance our understanding of how the JRVs adjust and reorganise at such scaled-up geometries, and answer the question of whether they are indeed the smallest superstructure units.

Quasistatic magnetoconvection of a fluid with low Prandtl number (${\textit {Pr}}=0.025$) with a vertical magnetic field is considered in a unit-aspect-ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number ${\textit {Ha}}$, the onset of convection is known to result from a sidewall instability giving rise to the wall-mode regime. Here, we carry out three-dimensional direct numerical simulations of unprecedented length to map out the parameter space at ${\textit {Ha}} = 200, 500, 1000$, varying the Rayleigh number (${\textit {Ra}}$) over the range $6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$. We track the development of stable equilibria produced by this primary instability, identifying bifurcations leading to limit cycles and eventually to chaotic dynamics. At ${\textit {Ha}}=200$, the steady wall-mode solution undergoes a symmetry-breaking bifurcation producing a state that features a coexistence between wall modes and a large-scale roll in the centre of the domain, which persists to higher ${\textit {Ra}}$. However, under a stronger magnetic field at ${\textit {Ha}}=1000$, the steady wall-mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in ${\textit {Ra}}$, the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of ${\textit {Ra}}$, and chaotic dynamics primarily arise from a mixture of chaotic wall-mode dynamics and arrays of cellular structures.

A scaling theory for the passive scalar transport in Couette flow, i.e. the flow between two parallel plates moving with different velocities, is proposed. This flow is determined by the bulk Reynolds number $Re_b$ and the Prandtl number $Pr$. In the turbulent regime, for moderate shear Reynolds number $Re_{\tau }$ and moderate $Pr$, we derive that the passive scalar transport characterised by the Nusselt number $Nu$ scales as $Nu \sim Pr^{1/2}Re_{\tau }^{2}Re_b^{-1}$. We then use the well-established scaling for the friction coefficient $C_f \sim Re_b^{-1/4}$ (corresponding to a shear Reynolds number $Re_{\tau } \sim Re_b^{7/8}$) which holds reasonably well within the range $3\times 10^{3} \leqslant Re_b \leqslant 10^{5}$, to obtain $Nu \sim Pr^{1/2}Re_b^{3/4}$ for the Nusselt number scaling. The theoretical results are tested against direct numerical simulations of Couette flows for the parameter ranges $81 \leqslant Re_b \leqslant 22361$ and $0.1 \leqslant Pr \leqslant 10$, finding good agreement. Analyses of the numerically obtained turbulent flow fields confirm logarithmic mean wall-parallel profiles of the velocity and the passive scalar in the inertial sublayer.

This work addresses the effects of different thermal sidewall boundary conditions on the formation of flow states and heat transport in two- and three-dimensional Rayleigh–Bénard convection (RBC) by means of direct numerical simulations and steady-state analysis for Rayleigh numbers ${\textit {Ra}}$ up to $4\times 10^{10}$ and Prandtl numbers ${\textit {Pr}}=0.1,1$ and $10$. We show that a linear temperature profile imposed at the conductive sidewall leads to a premature collapse of the single-roll state, whereas a sidewall maintained at a constant temperature enhances its stability. The collapse is caused by accelerated growth of the corner rolls with two distinct growth rate regimes determined by diffusion or convection for small or large ${\textit {Ra}}$, respectively. Above the collapse of the single-roll state, we find the emergence of a double-roll state in two-dimensional RBC and a double-toroidal state in three-dimensional cylindrical RBC. These states are most prominent in RBC with conductive sidewalls. The different states are reflected in the global heat transport, so that the different thermal conditions at the sidewall lead to significant differences in the Nusselt number for small to moderate ${\textit {Ra}}$. However, for larger ${\textit {Ra}}$, the heat transport and flow dynamics become increasingly alike for different sidewalls and are almost indistinguishable for ${\textit {Ra}}>10^{9}$. This suggests that the influence of imperfectly insulated sidewalls in RBC experiments is insignificant at very high ${\textit {Ra}}$ – provided that the mean sidewall temperature is controlled.

Using complementary experiments and direct numerical simulations, we study turbulent thermal convection of a liquid metal (Prandtl number $\textit {Pr}\approx 0.03$) in a box-shaped container, where two opposite square sidewalls are heated/cooled. The global response characteristics like the Nusselt number ${\textit {Nu}}$ and the Reynolds number $\textit {Re}$ collapse if the side height $L$ is used as the length scale rather than the distance $H$ between heated and cooled vertical plates. These results are obtained for various Rayleigh numbers $5\times 10^3\leq {\textit {Ra}}_H\leq 10^8$ (based on $H$) and the aspect ratios $L/H=1, 2, 3$ and $5$. Furthermore, we present a novel method to extract the wind-based Reynolds number, which works particularly well with the experimental Doppler-velocimetry measurements along vertical lines, regardless of their horizontal positions. The extraction method is based on the two-dimensional autocorrelation of the time–space data of the vertical velocity.

Thermally driven vertical convection (VC) – the flow in a box heated on one side and cooled on the other side, is investigated using direct numerical simulations with Rayleigh numbers over the wide range of $10^7\le Ra\le 10^{14}$ and a fixed Prandtl number $Pr=10$ in a two-dimensional convection cell with unit aspect ratio. It is found that the dependence of the mean vertical centre temperature gradient $S$ on $Ra$ shows three different regimes: in regime I ($Ra \lesssim 5\times 10^{10}$), $S$ is almost independent of $Ra$; in the newly identified regime II ($5\times 10^{10} \lesssim Ra \lesssim 10^{13}$), $S$ first increases with increasing $Ra$ (regime $\textrm {{II}}_a$), reaches its maximum and then decreases again (regime $\textrm {{II}}_b$); and in regime III ($Ra\gtrsim 10^{13}$), $S$ again becomes only weakly dependent on $Ra$, being slightly smaller than in regime I. The transition from regime I to regime II is related to the onset of unsteady flows arising from the ejection of plumes from the sidewall boundary layers. The maximum of $S$ occurs when these plumes are ejected over approximately half of the area (downstream) of the sidewalls. The onset of regime III is characterized by the appearance of layered structures near the top and bottom horizontal walls. The flow in regime III is characterized by a well-mixed bulk region owing to continuous ejection of plumes over large fractions of the sidewalls, and, as a result of the efficient mixing, the mean temperature gradient in the centre $S$ is smaller than that of regime I. In the three different regimes, significantly different flow organizations are identified: in regime I and regime $\textrm {{II}}_a$, the location of the maximal horizontal velocity is close to the top and bottom walls; however, in regime $\textrm {{II}}_b$ and regime III, banded zonal flow structures develop and the maximal horizontal velocity now is in the bulk region. The different flow organizations in the three regimes are also reflected in the scaling exponents in the effective power law scalings $Nu\sim Ra^\beta$ and $Re\sim Ra^\gamma$. Here, $Nu$ is the Nusselt number and $Re$ is the Reynolds number based on maximal vertical velocity (averaged over vertical direction). In regime I, the fitted scaling exponents ($\beta \approx 0.26$ and $\gamma \approx 0.51$) are in excellent agreement with the theoretical predictions of $\beta =1/4$ and $\gamma =1/2$ for laminar VC (Shishkina, Phys. Rev. E., vol. 93, 2016, 051102). However, in regimes II and III, $\beta$ increases to a value close to 1/3 and $\gamma$ decreases to a value close to 4/9. The stronger $Ra$ dependence of $Nu$ is related to the ejection of plumes and the larger local heat flux at the walls. The mean kinetic dissipation rate also shows different scaling relations with $Ra$ in the different regimes.

We present experimentally obtained time-averaged vertical temperature profiles $\theta (z)$ in horizontal convection (HC) in water (Prandtl number $Pr \simeq 6$), which were measured near the heating and cooling plates that are embedded in the bottom of HC samples. Three HC rectangular samples of different sizes but the same aspect ratio $\varGamma \equiv L:W:H = 10:1:1$ ($L$, $W$ and $H$ are the length, width and height of the sample, respectively) were used in the experiments, which allowed us to study HC in a Rayleigh-number range $2 \times 10^{10} \lesssim {Ra} \lesssim 9 \times 10^{12}$. The measurements revealed that above the cooling plate, the mean temperature profiles have a universal scaling form $\theta (z/\lambda _c)$ with $\lambda _c$ being a $Ra$-dependent thickness of the cold thermal boundary layer (BL). The $\theta (z/\lambda _c)$-profiles agree well with solutions to a laminar BL equation in HC, which is derived under assumption that the large-scale horizontal velocity achieves its maximum near the plate and vanishes in the bulk. Above the heating plate, the mean temperature field has a double-layer structure: in the lower layer, the $\theta$ profiles scale with the hot thermal BL thickness $\lambda _h$, while in the upper layer, they again scale with $\lambda _c$. Both scaling forms are in good agreement with the solutions to the BL equation with a proper parameter choice.

Recently, in Zhang et al. (Phys. Rev. Lett., vol. 124, 2020, 084505), it was found that, in rapidly rotating turbulent Rayleigh–Bénard convection in slender cylindrical containers (with diameter-to-height aspect ratio $\varGamma =1/2$) filled with a small-Prandtl-number fluid (${Pr}\approx 0.8$), the large-scale circulation is suppressed and a boundary zonal flow (BZF) develops near the sidewall, characterized by a bimodal probability density function of the temperature, cyclonic fluid motion and anticyclonic drift of the flow pattern (with respect to the rotating frame). This BZF carries a disproportionate amount (${>}60\,\%$) of the total heat transport for ${Pr} < 1$, but decreases rather abruptly for larger ${Pr}$ to approximately $35\,\%$. In this work, we show that the BZF is robust and appears in rapidly rotating turbulent Rayleigh–Bénard convection in containers of different $\varGamma$ and over a broad range of ${Pr}$ and ${Ra}$. Direct numerical simulations for Prandtl number $0.1 \leq {\textit {Pr}} \leq 12.3$, Rayleigh number $10^7 \leq {Ra} \leq 5\times 10^{9}$, inverse Ekman number $10^{5} \leq 1/{\textit {Ek}} \leq 10^{7}$ and $\varGamma = 1/3$, 1/2, 3/4, 1 and 2 show that the BZF width $\delta _0$ scales with the Rayleigh number ${Ra}$ and Ekman number ${\textit {Ek}}$ as $\delta _0/H \sim \varGamma ^{0} Pr^{\{-1/4, 0\}} {Ra}^{1/4} {\textit {Ek}}^{2/3}$ ($\{{\textit {Pr}}<1, {\textit {Pr}}>1\}$) and with the drift frequency scales as $\omega /\varOmega \sim \varGamma ^{0} Pr^{-4/3} {Ra}\,{\textit {Ek}}^{5/3}$, where $H$ is the cell height and $\varOmega$ the angular rotation rate. The mode number of the BZF is 1 for $\varGamma \lesssim 1$ and $2 \varGamma$ for $\varGamma = \{1,2\}$ independent of ${Ra}$ and ${Pr}$. The BZF is quite reminiscent of wall mode states in rotating convection.

This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in two-dimensional (2-D) and 3-D convective systems. For a broad range of Rayleigh numbers ($10^3\leq Ra \leq 10^7$) and thermal wave frequencies ($10^{-4}\leq \varOmega \leq 10^{0}$), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses and end up always travelling retrograde. In convection dominated regimes, however, mean flow advection, caused by tilted convection cells, becomes dominant. This generally leads to prograde directed mean zonal flows. By means of direct numerical simulations we validate theoretical predictions made for the diffusion dominated regime. Furthermore, we make use of the linear stability analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D simulations support the results for 2-D flows and thus provide further evidence for the relevance of the findings for geophysical and astrophysical systems.

Convection in a container, heated from below, cooled from above and rapidly rotated around a vertical axis, starts from its sidewall. When the imposed vertical temperature gradient is not sufficiently large for bulk modes to set in, thermal convection can start in the form of wall modes, which are observed near the sidewall as pairs of hot ascending and cold descending plumes that drift along the wall. With increasing temperature gradient, different wall and bulk modes occur and interact, leading finally to turbulence. A recent numerical study by Favier & Knobloch (J. Fluid Mech., 895, 2020, R1) reveals an extreme robustness of the wall states. They persist above the onset of bulk modes and turbulence, thereby relating them to the recently discovered boundary zonal flows in highly turbulent rotating thermal convection. More exciting is that the wall modes can be thought of as topologically protected states, as they are robust with respect to the sidewall shape. They stubbornly drift along the wall, following its contour, independent of geometric obstacles.

Classical and symmetrical horizontal convection is studied by means of direct numerical simulations for Rayleigh numbers $Ra$ up to 3 × 1012 and Prandtl numbers $Pr=0.1$, 1 and 10. For both set-ups, a very good agreement in global quantities with respect to heat and momentum transport is attained. Similar to Shishkina & Wagner (Phys. Rev. Lett., vol. 116, 2016, 024302), we find Nusselt number $Nu$ versus $Ra$ scaling transitions in a region $10^{8}\leqslant Ra\leqslant 10^{11}$. Above a critical $Ra$, the flow undergoes either a steady–oscillatory transition (small $Pr$) or a transition from steady state to a transient state with detaching plumes (large $Pr$). The onset of the oscillations takes place at $Ra\,Pr^{-1}\approx 5\times 10^{9}$ and the onset of detaching plumes at $Ra\,Pr^{5/4}\approx 9\times 10^{10}$. These onsets coincide with the onsets of scaling transitions.

Inclined turbulent thermal convection in liquid sodium is studied at large Rayleigh numbers $Ra\gtrsim 10^{7}$ based on the results of both experimental measurements and high-resolution numerical simulations. For a direct comparison, the considered system parameters are set to be similar: $Ra=1.67\times 10^{7}$ in the direct numerical simulations (DNS), $Ra=1.5\times 10^{7}$ in the large-eddy simulations and $Ra=1.42\times 10^{7}$ in the experiments, while the Prandtl number of liquid sodium is very small ($Pr\approx 0.009$). The cylindrical convection cell has an aspect ratio of one; one circular surface is heated, while the other one is cooled. Additionally, the cylinder is inclined with respect to gravity and the inclination angle varies from $\unicode[STIX]{x1D6FD}=0^{\circ }$, which corresponds to Rayleigh–Bénard convection (RBC), to $\unicode[STIX]{x1D6FD}=90^{\circ }$, as in a vertical convection (VC) set-up. Our study demonstrates quantitative agreement of the experimental and numerical results, in particular with respect to the global heat and momentum transport, temperature and velocity profiles, as well as the dynamics of the large-scale circulation (LSC). The DNS reveal that the twisting and sloshing of the LSC at small inclination angles periodically affects the instantaneous heat transport (up to $\pm 44\,\%$ of the mean heat transport). The twisted LSC is associated with a weak heat transport, while the sloshing mode that brings together the hot and cold streams of the LSC is associated with a strong heat transport. The experiments show that the heat transport scales as $Nu\sim Ra^{0.22}$ in both limiting cases (RBC and VC) for Rayleigh numbers around $Ra\approx 10^{7}$, while any inclination of the cell, $0<\unicode[STIX]{x1D6FD}\leqslant 90^{\circ }$, leads to an increase of $Nu$.

By means of three-dimensional direct numerical simulations, we investigate the influence of the regular roughness of heated and cooled plates on the mean heat transport in a cylindrical Rayleigh–Bénard convection cell of aspect ratio one. The roughness is introduced by a set of isothermal obstacles, which are attached to the plates and have a form of concentric rings of the same width. The considered Prandtl number $Pr$ equals 1, the Rayleigh number $Ra$ varies from $10^{6}$ to $10^{8}$, the number of rings on each plate is 1, 2, 4, 8 or 10, the height of the rings is varied from 1.5 % to 49 % of the cylinder height and the gap between the rings is varied from 1.5 % to 18.8 % of the cell diameter. Totally, 135 different cases are analysed. Direct numerical simulations show that with small $Ra$ and wide roughness rings, a small reduction of the mean heat transport (the Nusselt number $Nu$) is possible, but, in most cases, the presence of the heated and cooled obstacles generally leads to an increase of $Nu$, compared to the case of classical Rayleigh–Bénard convection with smooth plates. When the rings are very tall and the gaps between them are sufficiently wide, the effective mean heat flux can be several times larger than in the smooth case. For a fixed geometry of the obstacles, the scaling exponent in the $Nu$ versus $Ra$ scaling first increases with growing $Ra$ up to approximately 0.5, but then smoothly decreases back towards the exponent in the no-obstacle case.

In turbulent Rayleigh–Bénard (RB) convection with regular, mono-scale, surface roughness, the scaling exponent $\unicode[STIX]{x1D6FD}$ in the relationship between the Nusselt number $Nu$ and the Rayleigh number $Ra$, $Nu\sim Ra^{\unicode[STIX]{x1D6FD}}$ can be ${\approx}1/2$ locally, provided that $Ra$ is large enough to ensure that the thermal boundary layer thickness $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is comparable to the roughness height. However, at even larger $Ra$, $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ becomes thin enough to follow the irregular surface and $\unicode[STIX]{x1D6FD}$ saturates back to the value for smooth walls (Zhu et al., Phys. Rev. Lett., vol. 119, 2017, 154501). In this paper, we prevent this saturation by employing multiscale roughness. We perform direct numerical simulations of two-dimensional RB convection using an immersed boundary method to capture the rough plates. We find that, for rough boundaries that contain three distinct length scales, a scaling exponent of $\unicode[STIX]{x1D6FD}=0.49\pm 0.02$ can be sustained for at least three decades of $Ra$. The physical reason is that the threshold $Ra$ at which the scaling exponent $\unicode[STIX]{x1D6FD}$ saturates back to the smooth wall value is pushed to larger $Ra$, when the smaller roughness elements fully protrude through the thermal boundary layer. The multiscale roughness employed here may better resemble the irregular surfaces that are encountered in geophysical flows and in some industrial applications.

We critically analyse the different ways to evaluate the dependence of the Nusselt number ($\mathit{Nu}$) on the Rayleigh number ($\mathit{Ra}$) in measurements of the heat transport in turbulent Rayleigh–Bénard convection under general non-Oberbeck–Boussinesq conditions and show the sensitivity of this dependence to the choice of the reference temperature at which the fluid properties are evaluated. For the case when the fluid properties depend significantly on the temperature and any pressure dependence is insignificant we propose a method to estimate the centre temperature. The theoretical predictions show very good agreement with the Göttingen measurements by He et al. (New J. Phys., vol. 14, 2012, 063030). We further show too the values of the normalized heat transport $\mathit{Nu}/\mathit{Ra}^{1/3}$ are independent of whether they are evaluated in the whole convection cell or in the lower or upper part of the cell if the correct reference temperatures are used.

Any tilt of a Rayleigh–Bénard convection cell against gravity changes the global flow structure inside the cell, which leads to a change of the heat and momentum transport. Especially sensitive to the inclination angle is the heat transport in low-Prandtl-number fluids and confined geometries. The purpose of the present work is to investigate the global flow structure and its influence on the global heat transport in inclined convection in a cylindrical container of diameter-to-height aspect ratio $\unicode[STIX]{x1D6E4}=1/5$. The study is based on direct numerical simulations where two different Prandtl numbers $Pr=0.1$ and 1.0 are considered, while the Rayleigh number, $Ra$, ranges from $10^{6}$ to $10^{9}$. For each combination of $Ra$ and $Pr$, the inclination angle is varied between 0 and $\unicode[STIX]{x03C0}/2$. An optimal inclination angle of the convection cell, which provides the maximal global heat transport, is determined. For inclined convection we observe the formation of two system-sized plume columns, a hot and a cold one, that impinge on the opposite boundary layers. These are related to a strong increase in the heat transport.