Analytical and numerical investigations of the stability of laminar, fully developed pipe flow

The stability of fully developed laminar pipe flow is being reexamined to understand the start of the laminar-to-turbulent transition. While traditional theory predicts the flow remains stable for all Reynolds numbers (Re), experiments consistently show transition at finite Re. This discrepancy suggests that the theoretical analyses address a different, idealized flow than that seen in experiments. In this work, it is postulated that natural wall roughness, present in experiments but absent in theoretical treatments, accounts for this difference. To investigate this, the theoretical framework is extended to include flow in naturally rough walls, modeled as a porous medium. The same pressure gradient drives both the core flow and the flow within the rough-wall layer. A simple roughness model is first considered, allowing an analytical solution to capture the combined flow in the pipe and the roughness layer. To capture more realistic cases, a numerical approach is then developed for more complex roughness distributions. One such distribution is analyzed, and the results are compared with well-known experimental observations of pipe flow instability. A relationship for a critical pipe diameter Dc is derived, depending on the roughness, permeability, and thickness. The results indicate that low-pressure gradients, corresponding to low-Re flows, together with thin or highly permeable roughness layers, yield stable velocity profiles without inflection points. At higher Re, however, the velocity profiles develop inflection points, leading to the experimentally observed pipe flow instability, an effect of the flow in the roughness layer, previously neglected in theoretical treatments of laminar pipe flows. The final section presents additional results and outlines a framework for experimental validation of the numerical findings.