Low-order reaction-diffusion system approximates heat transfer and flow structure in annular convection

Abstract

Heat transfer in a fluid can be greatly enhanced by natural convection, giving rise to the nuanced relationship between the Nusselt number and Rayleigh number that has been a focus of modern fluid dynamics. Our work explores convection in an annular domain, where the geometry reinforces the large-scale circulatory flow pattern that is characteristic of natural convection. The flow must match the no-slip condition at the boundary, leading to a thin boundary layer where both the flow velocity and the temperature vary rapidly. To understand the system's heat transfer characteristics, we derive a reduced model from the Navier-Stokes-Boussinesq equations, whereby the equations of flow and heat are transformed to a system of low-order partial differential equations (PDEs) that take the form of a reaction-diffusion system. Solutions to the reaction-diffusion system, though they fail to predict dynamic events, preserve the same boundary-layer structure seen in the direct numerical simulation (DNS). By matching the solutions inside and outside the boundary layer, asymptotic analysis predicts a power-law relationship Nu $\propto$ Ra$^{1/4}$. Though difficult to distinguish from an exponent of 2/7, the predicted power law agrees well with measurements from DNS over several decades of the Rayleigh number. Considering the model's deficiencies in describing turbulent fluctuations and reversal events, the agreement regarding heat transfer characteristics is encouraging and suggests that the methodology of systematically deriving low-order PDEs from the governing equations may provide a useful complement to existing theories.