Laminar free convection induced by a line heat source, and heat transfer from wires at small Grashof numbers

The buoyancy-induced laminar flow and temperature fields associated with a line source of heat in an unbounded environment are described by numerically solving the non-dimensional Boussinesq equations with the appropriate boundary conditions. The solution is given for values of the Prandtl number, the single parameter, ranging from zero to infinity. The far-field form of the solution is well known, including a self-similar thermal plume above the source. The analytical description close to the source involves constants that must be evaluated with the numerical solution. These constants are used when calculating the free convection heat transfer from wires (or cylinders of non-circular shape) at small Grashof numbers. We find two regions in the flow field: an inner region, scaled with the radius of the wire, where the effects of convection can be neglected in first approximation, and an outer region where, also in first approximation, the flow and temperature fields are those due to a line source of heat. The cases of large and small Prandtl numbers are considered separately. There is good agreement between the Nusselt numbers given by the asymptotic analysis and by the numerical analysis, which we carry out for a wide range of Grashof numbers, extending to very small values the range of existing numerical results; there is also agreement with the existing correlations of the experimental results. A correlation expression is proposed for the relation between the Nusselt and Grashof numbers, based on the asymptotic forms of the relation for small and large Grashof numbers.