In this work, fully explicit numerical solutions for axial flow convection in cylindrical domains, completely or partially filled with a fluid saturated porous medium, are presented. Laminar natural convection in a vertical porous annulus and in a vertical annulus with a centrally located heat generating solid or porous rod, and laminar forced convection in a pipe partially filled with a porous medium are numerically simulated. Novel stability conditions are carried out for the axisymmetric version of the Artificial Compressibility (AC) Characteristic Based Split (CBS) algorithm, based on finite elements, in order to effectively solve the above problems. The results presented in this paper are validated against experimental and numerical data available from the literature. The problem taking into account the presence of the porous rod in the vertical annulus has been presented here for the first time. (C) 2012 Elsevier Ltd. All rights reserved.