Natural convection far downstream of a heat source on a solid wall

Abstract

An analysis is presented of some steady natural convection flows at large distances downstream of point heat sources on solid walls. These asymptotic self-similar flows depend only on the Prandtl number of the fluid. The flow induced by a localized source on an adiabatic wall that is vertical or facing downwards is described numerically, whereas the flow due to a localized source on a wall facing upwards separates and leads to a self-similar plume. When the wall is held at the same temperature as the ambient fluid far from the source, the flow is described by a self-similar solution of the second kind, with the algebraic decay of the temperature excess above the ambient temperature determined by a nonlinear eigenvalue problem. Numerical solutions of this problem are presented for two-dimensional and localized heat sources on a vertical wall, whereas the problem for a localized heat source under an inclined isothermal downwards-facing wall turns out to capture the Rayleigh–Taylor instability of the flow and could not be solved by the methods used in this paper. The analogous flows in fluid-saturated porous media are found to be the solutions of parameter-free problems. A conservation law similar to the one holding for a wall jet is found in the case of a two-dimensional source on an isothermal wall and numerical solutions are presented for the other cases.