Theoretical treatment of the instability of fully developed laminar pipe flow taking wall roughness into account

This paper concerns theoretical treatments of the stability of laminar, fully developed pipe flow, a long-standing problem in fluid mechanics that has not yet been solved. Numerous publications theoretically demonstrate that this flow is inherently stable to all types of small disturbances and across all Reynolds numbers (Re). However, apparently corresponding experiments show that laminar, fully developed pipe flows reach their state of transition at finite values of Re=(ŨD/ν), with Ũ= mean velocity, D= pipe diameter, ν= kinematic viscosity of fluid. An explanation for this “anomaly” is currently not available. The aim of this paper is to provide such an explanation by treating laminar pipe flows theoretically, taking wall roughness into account. This yields, for certain roughness properties and pipe diameters, a mean velocity distribution that possesses an inflection point and is, therefore, inherently unstable. A stability parameter, S=(4k0/δD), (k0= maximum permeability of wall roughness, δ= roughness layer thickness, D= pipe diameter) is derived that categorizes laminar, fully developed pipe flows into two classes: stable (when the parameter is >1.0) and unstable (when the parameter is <1.0). Hence, laminar pipe flows become unstable when the derived stability parameter approaches 1.0. Parallel carried out numerical studies confirm the results presented in this paper. One- and two-dimensional predictions yield similar results suggesting the stability parameter S be equal to 1.0 for a criterion for the onset of instability and transition in turbulence of laminar pipe flows. Nearly all available experimental results can be explained by the theoretical treatment described in this paper. An additional paper is in preparation to consider various results available on laminar pipe flows, such as region of constant Rec, dependence on pipe diameter and pipe length, and friction factor of laminar pipe flows.