On the Localized Transition of Pipe Poiseuille Flow, Part II: The Role of Tensile-Energy-Flux Vector

ABSTRACT: This work constitutes the second part of an investigation into the mechanical mechanism of laminar-turbulent transition in pipe Poiseuille flow, a canonical wall-bounded-flow problem in turbulence-research. Conventional hydrodynamic-stability studies mainly focus on disturbance effects while paying relatively little attention to the intrinsic mechanical properties of flow structures. In the present study, the tensile-energy-flux vector (renamed from the viscous-energy-flux vector in earlier work) and its divergence are theoretically derived and analyzed for pipe Poiseuille flow. The tensile-energy-flux vector vanishes both at the pipe wall and at the centerline of the pipe. Its divergence equals zero at the radial position 0.707R. Once the tensile force flow reaches its critical threshold value, this base-flow feature identifies 0.707R as a candidate radial location with enhanced susceptibility to localized laminar-turbulent transition. The combination of the critical-tensile-force-flow criterion and the critical-tensile-energy-flux criterion introduces a new critical-tensile-flux criterion. This predicted position lies within the radial oscillation range reported in published transition experiments. When the membrane theory for spherical shells is applied to the virtual fluid-shell element, this critical candidate position corresponds to a zenith angle of 45°. This angle bears a heuristic geometric analogy to the crack orientation observed in uniaxial-compression tests on rock specimens; nevertheless, the physical mechanisms governing solid fracture and hydrodynamic transition are fundamentally different. This paper further demonstrates that the critical-Reynolds-number criterion for pipe-flow laminar-turbulent transition is not an absolute universal constant. Under the working hypothesis that the critical-tensile-flux criterion behaves like a fluid-related property parameter, the ratio between critical tensile energy flux and cross-section-averaged kinetic energy flux exhibits an inverse-Reynolds-number scaling, reminiscent of the well-known inverse relationship between the laminar pipe-friction coefficient and Reynolds number.