We consider Rayleigh–Bénard convection in a layer of fluid between rough no-slip boundaries where the top and bottom boundary heights are functions of the horizontal coordinates with square-integrable gradients. We use the background method to derive an upper bound on the mean heat flux across the layer for all admissible boundary geometries. This flux, normalized by the temperature difference between the boundaries, can grow with the Rayleigh number ( $Ra$ ) no faster than $O(Ra^{1/2})$ as $Ra\rightarrow \infty$ . Our analysis yields a family of similar bounds, depending on how various estimates are tuned, but every version depends explicitly on the boundary geometry. In one version the coefficient of the $O(Ra^{1/2})$ leading term is $0.242+2.925\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$ , where $\Vert \unicode[STIX]{x1D735}h\Vert ^{2}$ is the mean squared magnitude of the boundary height gradients. Application to a particular geometry is illustrated for sinusoidal boundaries.

Convection of an internally heated fluid, confined between top and bottom plates of equal temperature, is studied by direct numerical simulation in two and three dimensions. The unstably stratified upper region drives convection that penetrates into the stably stratified lower region. The fraction of produced heat escaping across the bottom plate, which is one half without convection, initially decreases as convection strengthens. Entering the turbulent regime, this decrease reverses in two dimensions but continues monotonically in three dimensions. The mean fluid temperature, which grows proportionally to the heating rate ( $H$ ) without convection, grows proportionally to $H^{4/5}$ when convection is strong in both two and three dimensions. The ratio of the heating rate to the fluid temperature is likened to the Nusselt number of Rayleigh–Bénard convection. Simulations are reported for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in terms of the heating rate) up to $5\times 10^{10}$ .

We consider convection in an internally heated (IH) layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable ( $R_{L}$ ), and the energy analysis yields a Rayleigh number below which it is globally stable ( $R_{E}$ ). For various boundary conditions on the velocity, exact expressions for $R_{L}$ and $R_{E}$ are found using long-wavelength asymptotics. Each $R_{E}$ is strictly smaller than the corresponding $R_{L}$ but is within 1 %. The lower bound on the mean temperature is proven for no-slip velocity boundary conditions using the background method. The bound guarantees that the mean temperature of the fluid, relative to that of the top boundary, grows with the heating rate ( $H$ ) no slower than $H^{2/3}$ .

We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ( $\mathit{Pr}$ ) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ( $\mathit{Ra}$ ) sufficiently, and we explore the resulting convection for $\mathit{Ra}$ up to $10^{10}$ . When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $\mathit{Ra}\rightarrow \infty$ . The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $\mathit{Ra}$ . When the large-scale shear is present with $\mathit{Pr}\lesssim 2$ , the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $\mathit{Ra}$ for $\mathit{Pr}=1$ . When the shear is present with $\mathit{Pr}\gtrsim 3$ , the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $\mathit{Ra}$ , but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to $\mathit{Ra}^{0.077}$ when $\mathit{Pr}=3$ and to $\mathit{Ra}^{0.19}$ when $\mathit{Pr}=10$ . Analogies with tokamak plasmas are described.