Sidewall effects in Rayleigh-B\'enard convection

We investigate the influence of the temperature boundary conditions at the sidewall on the heat transport in Rayleigh-B\'enard (RB) convection using direct numerical simulations. For relatively low Rayleigh numbers Ra the heat transport is higher when the sidewall is isothermal, kept at a temperature $T_c+\Delta/2$ (where $\Delta$ is the temperature difference between the horizontal plates and $T_c$ the temperature of the cold plate), than when the sidewall is adiabatic. The reason is that in the former case part of the heat current avoids the thermal resistance of the fluid layer by escaping through the sidewall that acts as a short-circuit. For higher Ra the bulk becomes more isothermal and this reduces the heat current through the sidewall. Therefore the heat flux in a cell with an isothermal sidewall converges to the value obtained with an adiabatic sidewall for high enough Ra ($\simeq 10^{10}$). However, when the sidewall temperature deviates from $T_c+\Delta/2$ the heat transport at the bottom and top plates is different from the value obtained using an adiabatic sidewall. In this case the difference does not decrease with increasing Ra thus indicating that the ambient temperature of the experimental apparatus can influence the heat transfer. A similar behavior is observed when only a very small sidewall region close to the horizontal plates is kept isothermal, while the rest of the sidewall is adiabatic. The reason is that in the region closest to the horizontal plates the temperature difference between the fluid and the sidewall is highest. This suggests that one should be careful with the placement of thermal shields outside the fluid sample to minimize spurious heat currents.