Effect of curvature on transient natural convection in a vertical circular pipe

Natural convection adjacent to a curved vertical wall is widely present. Unfortunately, the effect of curvature on the transient thermal boundary layer (TBL) adjacent to the concave vertical wall has been neglected. In this study, dynamical evolution and thermal process of transient natural convection in a vertical circular pipe are discussed using scaling analysis, a boundary flow regime for the thin TBL without merging and a duct flow regime for the TBL with merging at the axis of the pipe are distinguished. The scaling laws quantifying the dependence of thickness, velocity and flow rate of the TBL of the fluid with the fixed Prandtl number in the vertical pipe on the Rayleigh number (Ra-T and Ra-q) and the ratio of height to radius of the pipe (A) are first reported for the isothermal and isoflux conditions. The curvature effect becomes stronger with the increase of the thickness of the TBL. Under the duct flow regime, the non-dimensional flow rate is scaled with Ra-T(1/2) A(-1) for the isothermal condition and with Raq(1/2) A(-3/2) for the isoflux condition. The scaling laws of the thickness, velocity and the flow rate of the TBL in the vertical pipe are validated based on the numerical results from direct numerical simulation (DNS) with good precision. The scaling coefficient is also presented under different regimes and conditions, which can serve as a design guide to determine natural convection in the vertical circular pipe.