Combined convection flow in a vertical duct with wall temperatures that vary linearly with depth

Abstract

Numerical solutions are presented for the problem of steady laminar combined convection flows in vertical parallel-plate ducts, with linearly varying wall temperatures. Neglecting streamwise diffusion in the analysis leads to a parabolic set of governing equations. These are solved using a marching technique for an implicit finite-difference scheme with vorticity, streamfunction, and temperature as independent variables. Various values of the governing parameter, the Grashof number, Gr, are considered, including the forced convection solution, Gr = 0, while the Prandtl number, Pr, is set at a value of unity in order to present the numerical method. As the value of |Gr| increases, reverse flow regions appear that are present in the fully-developed flow; these are dealt with using a modification of the standard marching technique. Results are obtained in terms of velocity profiles, local Nusselt numbers, flow average temperatures, and friction factors, and the comparative strengths of the recirculation regions are assessed. A simple correlation is given for the development lengths, in terms of Gr, for Pr = 1.