Heat transfer and large-scale circulation in annular convection

The heat transfer of fluid can be greatly enhanced by natural convection, leading to the famous Nu-Ra scaling that has been a focus of modern fluid dynamics. Our work explores natural convection in an annular domain, where the annular geometry reinforces the large-scale circulation (LSC) and leads to an altered Nu-Ra scaling. To understand the heat transfer and flow pattern in this novel geometry, we derive a reduced model from the Navier-Stokes-Boussinesq equations where the equations of flow and heat are transformed to a system of low-order reaction diffusion equations. As we increase the Rayleigh number, this reduced model recovers the three states seen in the direct numerical simulation (DNS): the motionless conductive state, the circulating state with a steady LSC, and the reversal state where the LSC reverses direction spontaneously. Moreover, the reaction diffusion equations preserve the same boundary layer structures seen in the DNS, allowing us to directly measure the Nusselt and Reynolds numbers. By matching the solutions inside and outside the boundary layer, we recover the Nu-Ra and Re-Ra scaling law of the DNS, further demonstrating the accuracy of this reduced model. Our results also provide a systematic way of analyzing thermal convection in an annular domain, which brings us one step closer to understanding the origin of LSC and the mechanism of convective heat transfer.