On the Localized Transition of Pipe Poiseuille Flow. Part I: The Role of Tensile Force Flow

The early turbulence phenomenon has been observed in pipe flow of very dilute polymer solutions [1–4], and the full chord laminar flow can be achieved on various laminar suction wings at high Reynolds numbers (Re) up to approximately 2 000 000 [5–7]. Their transition conditions deviate significantly from the traditional criteria, critical Re about 2 000~2 300, which is quoted in most contemporary textbooks for pipe flow [8–13]. In this paper, a new force model with a virtual fluid layer, which is of a hemispherical shell shape and with a constant thickness inside a laminar pipe flow is established, on the basis of the membrane force model of a spherical shell under uniformly distributed load conditions in structural mechanics. In laminar flow state with a lower Re and a lower pressure gradient, the curvature radius of the virtual spherical liquid layer is inversely proportional to the pressure gradient. As Re increases, pressure gradient also increases, while the curvature radius decreases. When the curvature radius decreases to be equal to and starting less than the pipe radius, the stable liquid layer structure collapses, and the laminar flow becomes turbulent. This is a transition state with a critical tensile force flow defined as twice the product of the viscosity of the fluid and the maximum velocity in pipe, divided by the pipe radius. In laminar flow situation, the shear stress at the pipe wall can be interpreted as a horizontal component of the critical tensile force flow, and the direction is against the flow. Only when the flow achieving the transition condition, the shear stress at the wall become the critical tensile force flow itself, which had already been observed in early turbulence [1–4,14]. The second case, which can be explained by the concept of critical tensile force flow, is high Re laminar pipe flow [5–7], for example, the pipe with surface suction can be considered as a part of a virtual, larger pipe with a no slip wall at where the shear stress coincides with the critical tensile force flow, the shear stress at the real pipe is smaller, with a weakening factor related to the ratio of the average velocity in the real pipe to its maximum velocity.