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Heat transfer in a fluid can be 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. Within such a geometry, a novel Nusselt-Rayleigh scaling emerges, suggesting subtle differences in the heat transfer mechanisms as compared to the well-known case of Rayleigh-B\'enard convection. To understand the heat transfer characteristics of this system, 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 partial differential equations (PDEs). This system of PDEs takes the form of a reaction-diffusion system, and its solution preserves the same boundary layer structures seen in the direct numerical simulation (DNS). By matching the solutions inside and outside the boundary layer, our asymptotic analysis recovers the Nusselt-Rayleigh relationship measured in DNS and yields a power-law scaling with exponent 1/4. |
